I.

Well, this title reveals nothing new:

Most mathematicians think of their subject as an art. This being not enough for some, they actually attempt to produce (sometimes dubious) art. And, on the other side, many artists show an inclination to either geometry or to an axiomatization of their findings and to deductive analytic reasoning. Some even explicitly like (usually school-level) mathematics.

But...

although these feelings and connections are individually profound it is surprising that often they are interpreted superficially, just from the observer's point of view. What we want to say is that these mathematical interpretations of art works concentrate on the formal aspects of artist's work which on the mathematical side are easy, sometimes banal, and never up-to-date.

On the other hand the artistic interpretations of mathematics are mostly not even serious (or they concentrate on general questions of "aesthetic" qualities and criteria of mathematical work).

II.

Why? Why so easily? Is there nothing deeper to be said here?

Do not Giacometti's beautiful line drawings and portraits (Figure 1) resemble the beauty of complicated geometrical constructions (Figure 1)? In both cases the seeming randomness of their separate parts gives a compact global information and perhaps a surprising impact of the whole⁽¹⁾ .

Does not a "mathematical" sketch⁽²⁾ (Figure 2) shows striking similarities to a Gončarova classical sketch (Figure 2)?

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Figure 1 ... Giacometti's line drawings resemble the beauty of geometrical constructions ...

Figure 2 ... a "mathematical" sketch shows similarities to a Gončarova classical sketch ...

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Do not the "random" paperworks of Beuys' and his drawings searching deep into the unconsciousness, resemble the preparatory sketches of a scientist mirroring the tortuous search for a missing link or pattern (Figure 3)?

Does not a late Max Ernst print share qualities with a David Hilbert sketch from his Grundlagen der Geometrie [Hi] (Figure 4).

These examples are by no means the only ones. Perhaps they are not even representative for our purpose. They are intended merely to illustrate our initial position. But they should be compared, for example, with the arbitrariness of the approaches of authors of the XLII Bienale di Venezia where, under title Arte e Scienza, the organizers touched just the surface - obscuring the profound connections to the formally unifying principle of the whole exposition. (But of course fulfilling some other goals.)

III.

The formal similarities between art and various aspects of mathematics (mostly geometry) are both classical and contemporary.

The mysticism of the golden ratio, of the pentagon and the regular polyhedra, of the complicated mathematical questions of perspective and of vision in general, all this has intrigued both artists and scientists for centuries. From numerous book let us refer to [B],[Zh] and the more recent collection [S2]; but we shall not pursue this direction. Mathematics was always a source of complex (for outsiders, mystical) patterns. This is even more true today than ever. With our understanding of randomness and aided by graphical devices attached to computers, this "artistic potential" of mathematics is growing. It is our conviction that this pattern generation displays mostly formal artistic similarities. However intriguing this may be, for example Escher's work displays a dizzying mastery this has no bearing on the mainstream of art, on the main problems which art and

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artists have solved and are solving today. And in a sense this is based on combinatorially refined byproducts of mathematics as well.

One should beware of simplistic analogies and misleading terminology: The so-called +_ Mondrian style has as much bearing on mathematics as the Art Gallery Problem has on art or the Piano-Mover Problem on music⁽³⁾.

IV.

What then is our main thesis?

We would like to rephrase the words Kandinsky used when comparing art and music:

In our opinion the similarities of art and mathematics are evident but they lie very deep.

These similarities are best traced by considering together the development, the creative activity and the methods of modern mathematics and of modern art. Some of these we are going to list, two of them we shall cover in greater detail.

Rather than considering the final products (the work of art or theorems) we wish to stress the similarities in the modus operandi of artists and mathematicians.

We wish to stress that some of these similarities can be traced to all levels and moments of the production of both art and mathematics. We are convinced that these similarities stem mainly from the idealistic character of both pursuits.

This does not contradict the fact on which we all agree that in both cases the handicraft, namely the refined routine part is both essential and difficult and only after understanding it can the actual work start. Art academies produce as few artists as universities produce new mathematicians⁽⁴⁾.

But again...

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Figure 5 ... a sketch due to Leibnitz ...

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Our similarities, however intuitively obvious they may be, are difficult to document. There are great social, formal and practical differences between "scientific" and "artistic" methods and approaches. Thus our unifying principle is work - the producing activity itself.

Our standpoint is that of a workaholic.

V.

Our first example is the activity of sketching and its product, sketches. This is a very well defined artistic category. A sketch is usually an object of simple form, mostly paperwork and mostly drawing (sketches in other media are quite rare). Artists are educated to do sketches, everybody can buy a "sketchbook" and artists frequently do sketches. Some even claim that one should sketch every day.

Artists consider sketches as their private diaries and consequently do not like to part with them. Sketches, not to mention sketchbook, are thus considered by artists as very personal property. They are valuable collectibles and some have even reached facsimiled and commented editions (see e.g. [De], [Ce] and most great masters; but also [Ma] and [Ti]⁽⁵⁾).

Why do we regard these preliminary works so highly? Is their market value (rarity) and (sometimes) their obviously raw skill the only reason?

In our opinion the main reason is that by means of them we can understand better the genesis of particular works. A sketch may be a direct witness to the latent power and the artistic tendency of an artist. And perhaps best, quoting Bernhard Korte, a sketch presents an encoding of a fantasy.

By means of sketches we can understand more fully the crucial ideas and motivation of artists. Ideas which helped to create modern art history.

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What then are mathematical sketches?

What is the analogy of preliminary sketches and necessarily incomplete works in an exact and precise discipline like mathematics? Are these "private communications" or modern means of communications like "preprints" or the more recent "E-mail messages"?

None of these. Even personal letters are usually formal. This is in a sense necessary and caused by the very nature of mathematics. Mathematicians live a paradox: although pure ideas are crucial for their work (and the whole subject has a developed philosophy) these ideas mean nothing without (absolutely) true statements and (rigorous) proofs. (The later two usually being highly technical.) For speculation (in the written form) there is no place⁽⁶⁾.

This in turn leads to the fact that mathematicians hide their original ideas and (true) motivations. They tend to write in the well known dry Definition - Theorem -- Proof style (great German mathematics of the 19th century is probably largely responsible for this)⁽⁷⁾.

What then are the sketches of mathematicians?

Yes, mathematicians do sketches, in fact many of them. And they value them and they usually hide them. It is not easy to see them, they are made in perishable ink on small pieces of paper.

Where, and better how, can we then see them?

Well, we can see them when observing somebody working. And to make things even more difficult, the usual condition is that we work jointly with the person. An outside observer may destroy or disturb the actual moment and gem of a mathematical discovery.

In order to be more specific, allow me at this moment a personal story. I was working once with one of the grand old men⁽⁸⁾ of mathematics. We were trying to solve a problem which was related to the size of an unknown set.

The old man said "Perhaps the set is non-empty" and drew a suspicious picture similar to the Figure 6.

After a while he said: "Perhaps the set is finite" and he drew Figure 7.

And then after a while: "I don't know, perhaps it is even infinite" which he visualized by⁽⁹⁾ Figure 8.

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Figure 6 The old man said: "Perhaps it is non-empty..."	Figure 7 than he said: "Perhaps is finite..."	Figure 8 and then after a while: "I don't know, perhaps it is even..."
	or
or
	or
Figure 9 ... you may chose any picture ...

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Such is the role of personal contact in mathematics, the role of research seminars, the role of problem seminars. The possibility to see mathematicians at work, to see them do just sketches. And this is a very rare opportunity which (up to now) is not even possible to buy.

Mathematicians also use sketches to visualize their thinking even (and preferably) in areas which cannot be visualized.

It matters how you visualize (an n-dimensional) cube. You may chose any of the following five pictures (Figure 9).

In a sense, all these representations of a cube are equivalent, yet they relate to different mathematical disciplines. These particular approaches are represented, sketched and in fact encoded by these pictures. These sketches pave the road to the work itself. Again, they are encodings of a fantasy.

Well artists have more possibilities and their cubes may take more adventurous forms. We decided to include Jiří Načeradsky's sketches for one of his "chinese boxes" - La Chose Enigme (Figure 10).

One can go on and describe other common features of mathematical and art - sketches. Let us add just one remark pertinent to both fields.

A common thing is that good sketches carry ideas - patterns of thoughts and forms. But these ideas are sometimes difficult to recognize. And per se the sketches often have no value. What is the value of Cézanne's sketch except that it was done by Cézanne and that it is related to some of his crucial works?

What is the value of a sheet of paper with a cube and ball on it?

However the informed professional (who is working jointly) may recognize the idea, may recognize the crucial break. This intimate character of sketches was responsible for some of the major influences and controversies in both artistic and scientific history ⁽¹⁰⁾.

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VI.

Let us now turn our attention to finished works: paintings, sculptures, objects on one side and ... - what belongs here on the other side?

(Long difficult) articles in (prestigious) journals? Influential or difficult (possibly unreadable) books?

None of these.

A finished (theoretical) mathematical work is an idea. But an idea which is justified (sometimes tediously) and verified (usually in a precise and technical manner). And although this may seem to be static and permanent it is actually very dynamic. The place which a particular idea (work) occupies depends on how it influences other works and in which actions it takes part. The value of a work is derived from the activity surrounding it (for example the particular activity that "one cannot improve it").

It is well known that the predominance of ideas over the material side of the subject is nowhere demonstrated so convincingly as in mathematics⁽¹¹⁾.

On the other hand the relationship between the idealistic and the formalistic aspects of artistic works is very subtle. It is one of the classical and most frequently discussed topics of art history. This is of course beyond the scope of our essay. Let us just mention a few relevant points.

There is evidence that with time the ideal part of an artistic work seems to be growing. Modern art is transcending (no doubt with the help of other media) descriptive motivation and is transforming itself to more idealistic and more dynamic levels. We shall comment on this later and in greater detail.

In some recent forms the ideas seem to be absolutely predominant. (Some might claim that too much so, that the works lack technical skills and display a suspicious absence of mastery.) We are not referring here to "action painting" and such forms. What we have in mind is that line of modern painting which comes from Max Ernst's beautiful program line "Ein Magier zu werden und den Mythos

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seiner Zeit zu finden". We think here of many of the actions of fluxus and the happenings of the sixties and seventies which in retrospect have perhaps only one value and that is pure ideal: these actions turned our attention to new areas and changed our sensibility.

So it was when Beuys collected public garbage and exhibited it, so it was when he came up with his project of planting trees. With a slight exaggeration he was conveying to a small but attentive public as profound a message as one conveys by deciding to start a new scientific seminar. As if one says to a (of course very small) group of talented young students "I like this and I would be happy to work with you on it". But the well balanced words and gestures have to correspond to a well balanced idea and mind. Great things may have simple and humble origins⁽¹²⁾.

Perhaps nowhere has modern art found itself so close to mathematics as in the time of its various minimalistic programs. This is the second area which we want to discuss in greater detail. This time we shall start with a brief description of the mathematical side.

VII.

The duality of synthetic and analytical thinking is inherent in most human activities. In one of its purest forms it resides in mathematics in the discipline whichprides itself by giving and analyzing "Laws and Thoughts".

The role of deduction and synthesis, the role of derivation and axiomatization, was discovered in mathematics. The first time it was presented to the world was perhaps in Euclid's Elements. This influenced all the sciences as an ideal norm.

However it is perhaps fair to say that only the new and modern concepts of the second half of 19th century freed our mathematical imagination and the amazing structural variety of mathematics of the twentieth century was created. The creation and use of very abstract and, at their time of creation, seemingly meaningless notions - such as set, relation, algebraic and logical system - led to a dazzling variety of new theories and programs. Whole new disciplines were created. They were not created in a speculative way but naturally in a pursuit of new horizons, in solving fundamental

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problems, by way of analyzing their roots. Diversity and abstraction led to natural questions about simplifications, about the foundations and axiomatization of various branches of mathematics. This reductionism was very radical and for several decades it seemed that all mathematical activity reduced to the pursuit of simple logical rules, that it suffices to apply logical rules to a suitably selected group of primitive notions and axioms. This was an explicit program of many practitioners and of several leading mathematicians (most notably Russell and Hilbert). This period is sometimes described as the search for foundations. This is slightly misleading because it was not a passive search for something known but rather a journey into an unknown land.

And this search yielded as a byproduct several cornerstone notions of modern mathematics: the development of set theory, model theory, the definition of an abstract computing machine, the understanding of the notion of an algorithm. These notions were the precursors of actual products, concrete things. And at these dramatic periods the very best people were involved in those activities. These comments are by no means restricted to mathematical logic only. They apply to, say, the prewar development of topology as well (see e.g. [Wh]).

This radical program ended in catastrophe, with the collapse of the whole project. It is the nature of mathematics that a few key results by Gödel and Church were sufficient for this débacle: there is no algorithm to solve certain naturally posed problems and every axiomatization of a non-trivial theory contains true statements which one cannot deduce from that given set of axioms.

The impossibility of a logical reduction of any meaningful theory had a profound influence on the philosophy of mathematics, on the way in which

mathematicians perceived their art. Of course it influenced less the mainstream of mathematical activity, especially after these results were well understood. The notions and the philosophy were absorbed into general mathematical maturity, and attention was turned elsewhere⁽¹³⁾. Why? Again: The entire area and its main principles became well understood.

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VIII.

Mathematics is not a single activity with a spectacular development during that amazing 100 year period divided nearly equally into two centuries. On the contrary, as in most human activities this was the period of a rapid and a decisive growth which profoundly shaped our present.

Art is no exception to these general comments. For our purpose we briefly outline some of the relevant features of its history. They should be compared with remarks stated above for mathematics.

In the second half of 19th century art freed itself from the burden of the ultimate criterion of likeness to reality and, through the courage of its leading figures, from the burden of order. Thus freed, art progressed extremely quickly. This development rightly deserves to be called a revolution. From Cézanne's crucial discoveries to the definitive abstract works of postcubism took less than 10 years⁽¹⁴⁾.

It seems that despite all the divergences the mainstream went in search of the essence of art which would definitely free works from the ballast of conventions and irrelevant findings.

This was done partly by Cézanne's reductions and partly by van Gogh and others. The destruction of old forms was successful and complete. It ended (only) a few years later with abstract forms, with the new artistic vocabulary of Klee, Picasso, Kandinsky, Kupka, Mondrian, the Russians and others. This elementarism was a natural state of affairs after the completion of the previous stages.

Consider the famous and well documented "Avignon-path" of Picasso. Consider the equally famous "Tree-path" of Mondrian. Or the less known mystical path taken by Kupka to his first geometrical - abstract drawings - his now famous "plans":

Then consider the parallel developments in the mainstream of modern mathematics, its struggle for the right definitions and for the right notions which would capture (and free) our imagination⁽¹⁵⁾.

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Perhaps these adventures, the richness of the development, led to the belief that the essential symbols and principles were already discovered, that art in future would be derived by applying principles .... The proud mind was triumphant (under various names suprematism, constructivism, rayonism, neoplasticism):

"... now when we do not care about things themselves we can construct the picture as follows ..." (M. Larionov)

"...for a long time I looked for individual forms and natural colours which inspire subjective feelings and hide pure reality ... it is necessary to reduce the natural forms all the way to the pure fixed ratio" (P. Mondrian)

"...constructivism proves that one cannot determine the border line between mathematics and art... Modern art obtained pure intuitively and independently the same results as the modern science ..." (El Lisickij)

"...reducing one detail after another we see the basis, the principle of construction consists of a few large planes..." (F. Kupka)

"...a system which creates itself in time and space, independent of notions of aesthetic beauty ..." (K. Malevič)

"... I found similarities between art and chess-playing ..." (M. Duchamp)⁽¹⁶⁾

To make the parallel with mathematics complete one has to only remark that these projects, not the art itself, collapsed some years later. There was no Gödel in art and this was also not necessary. The (very) restrictive views of various minimalistic-constructivistic theories collapsed either by themselves (having lost their momentum) or by changes in the society which turned its attention to broader and less rigid art.

Note that this collapse was related to the programs and not to the works themselves. This is another feature that mathematics shares with art: the stellar permanence of its true findings. The maestros of international gothic are not outdated and we somehow feel they never will be. Their pure, limited means and the golden-blue

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nobility speak to us with the same intensity as do ancient artefacts, in the same way as the ancient lore of Euclid and other great masters of the past.

IX.

Let us summarize.

Our intention in this essay has been to draw some parallels between art and mathematics from the creative point of view. We have tried to stress some similarities in the creative processes of both areas. Admittedly, by so doing we neglected the more common approach of simply comparing the results of these processes. But the few illustrations which we included are complemented by many formal similarities of art and chip design in the rest of this catalogue.

The interested reader probably knows many other examples of such similarities from his own experience, examples from various type of activities.

Is the existence of such similarities just a coincidence? Is it just one of God's whims? Is it just one of man's fabrications?

We do not know ...

Let us end this essay in a highly speculative way:

In mathematics there is a principle called "Church's thesis" which in essence claims that every (intuitive) algorithmic procedure takes a particular form, a form expressed by (any) universal computing machine.

Maybe something like Church's thesis holds in general for human activities. Let us try to formulate this thesis as follows:

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All sufficiently deep activities, all sufficiently deep understandings have profound similarities. This is exhibited in the way the work (knowledge) is organized, in the way it is revealed and in the way it interacts with other activities.

Admittedly this "Creative Thesis" is too vague. But one has to expect this at such a level of generality.

Perhaps one should interpret the thesis positively: How often are stressed only the conflicting features of artistic and scientific communities, say, along the lines of feelings and sensibilities as opposed to mind and brain activities. We feel that these views are often superficial and isolationist, and sometimes simply an overreaction.

By the same token it is difficult to find a counterexample to the thesis. Mathematics, in particular, is a rich generator of complex patterns and as such it is often used. So it will be difficult to break the mathematical trap. But also in the area of artistic activities one can speculate that out of efforts to break the similarities were found forms like surrealism, automatic writing and the like. We would like to say that these were searches for a counterexample to the Creative Thesis. And these cases were unsuccessful searches.

Chip design represents one of the most concentrated of human activities. This activity is interdisciplinary with methods spanning computer science, mathematics, physics and even philosophy. It is no wonder that the interplay of these activities displays some affinity to artistic works. The degree of these similarities is striking; this also supports our thesis and is the main purpose of this book.

Acknowledgements

In spite of its sketchy character this essay is a result of longstanding interests and activity. Perhaps there will be an opportunity to write a more complete version. I benefited from many discussions with many (both mathematical and artistic) friends. Earlier versions were published by Institut für Discrete Mathematik, Universität Bonn and a german translation by Fakultät für Mathematik, Universität Bielefeld. This is not the place to thank them all. But part of my ambition was to illustrate this essay by works of

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contemporary Czech artists: Jan Smetana (born 1918), Karel Marx (born 1928), Jiří Načeradsky (born 1939) and Ivan Ouhel (born 1945) - all living and working in Prague. I thank them for lending me their works.

The sketches in the headlines of individual sections are then based on the work of the following artists: I. Ch. Payan (Grenoble), II. P. Klee, III. Le Corbusier, IV., VI., VIII. J. Načeradsky, IX. Jan Smetana. The headlines of Section V contain two sketches of Leibnitz (kept in Niedersächsische Landesbibliothek Hannover) together with a contemporary sketch which I discovered on a blackboard in nearby Bielefeld. Karel Marx is the author of a sketch this page. Page contains a copy of a sketch due to Leibnitz and (Figure 20) a copy of a page from perhaps the most famous mathematical sketchbook - Mathematical Diary of Gauss. Authors of other sketches are introduced in the text.

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References

[Be] Beuys vor Beuys DuMont, Köln 1987

[Bu] M. Bunim: Space in Medieval Painting and the Forerunners at Perspective, Columbia University Press, New York, 1940

[De] Degas: A sketchbook, Dover, New York, 1988

[Ga] C.F. Gauss: Mathematisches Tagebuch 1796-1814, Geest & Portig, Leipzig 1976

[Hi] D. Hilbert: Grundlagen der Geometrie, Teubner, 1934

[Kl] P. Klee: Pedagogical Sketchbook, Faber and Faber, 1953

[Ka] W. Kandinsky: Punkt und Linie zu Fläche, Bauhaus Books (ed. W. Gropius, L. Moholy-Nagy), reprinted Dover 1979

[Ku] M. Nešlehová: B. Kubišta, Odeon 1984

[L] I. Lakatos: Proofs and Refutation, Cambridge University Press, 1976

[La] M. Lamač: Thoughts of modern artists, Odeon, Praha, 1989

[Le] G.W. Leibniz: (Katalog), Universität Hannover 1990

[Ma] H. von Mareé: Skizzenbuch, Von der Heydt-Museum, Wuppertal, 1987

[Po] G. Polya: How to Solve It, Princeton University Press, 1945

[S2] Symmetry 2 (ed. I. Hargittai), Pergamon Press, 1989

[Ti] M. Šafránek: French years of František Tichy, NČSVU, Praha 1965

[Wh] H. Whitney: Moscow 1935: Topology moving Toward America. In: A Century of Mathematics in America, AMS 1988, pp. 97-118.

[Zh] L.F.Zhegin: The Language of Painting, Iskustvo, Moscow 1970

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1. There is more to this statement than meets the eye. Geometrical configurations have simple local rules and yet have a complex global structure (rigidity). This has an important (unpleasant) consequence for a computer handling geometrical questions. Technically: the trees for search strategies tend to have large depth.

2. Based on a photograph by Adrian Bondy

3. See: O'Rourke: The Art Gallery Problem, Oxford University Press, Oxford

M. Sharir, J. Schwarz: Piano mover problem I.- X., Courant Institute of Mathematical Sciences, New York

4. But as remarked by Dominic Welsh "it is much harder to get into art school than to do mathematics at university".

5. (Tichy was a foremost Czech avant-garde artist.) In Tichy one can find an emotional testimony to the intimate value of sketchbook: "Tichy valued these sketches and unfinished studies and the whole Marseille-sketchbook more than his finished works. For an artist, as for an inventor, the initial pleasure is replaced by indifference and even repulsion, which was often Tichy's case. Tichy gave me this sketchbook as a very special gift."

6. This is not a mathematical text.

7. Only rarely this style is broken. One may quote for example [Po], [La]. However these are textbook-type reports which inform us a posteriori about the adventures of creation. In this way these texts should be compared with texts like [Ka], [Kl]. However enchanting they may be, they report to us a sterile distillation of their authors' powers.

8. P. Erdös

9. This story may sound too simplistic. So let me complement it by the following: A few years ago I was giving a talk for Prague highschool students with the intention of encouraging them to study mathematics at Charles University. I wanted to use the same story. I was discouraged by my colleagues on the basis that the example is too complicated and deep.

10. Did Tatlin see Picasso's guitar? How much did Einstein know about Poincaré? Did Leibnitz know about Newton? And how much did König know about Probenius? How much does the UNO building owe to a Corbusier sketch?

11. At the most common level, mathematicians all over the world are proud that they need just blackboard, chalk, pen and paper (and these days E-mail).

12. It is beside the question that the truly great fluxus artists produced art in the classical sense as well. An example for all was recently presented by Jim Dine (Ny Carlsberg Glyptotek, July 1990).

13. This comment should be applied to the mainstream of that development of logic which was motivated by Hilbert's program only. Some of the later highlights include solutions of Hilbert's problems by Cohen and Matijasevič, and the mathematically interesting version of Gödel's incompleteness theorem by Paris and Harrington.

14. For some artists the given time was incredibly short. This is the case of Macke, Appollinaire and even more so for Kubišta [Ku] who absorbed the whole development in merely two and half years.

15. Once logical deductions are mastered, ideas are carried by definitions.

16. These (admittedly too sketchy) lines are extracted from a beautiful book by M. Lamač, "Thoughts of modern artists" Odeon, Praha 1989

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