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The Meaning of Beauty in Exact Natural Science


Lecture delivered to the Bayerische Akademie der Schönen Künste

When a representative of natural science has to address a session of the Academy of the Fine Arts, he can hardly dare to express opinions about the topic art. For the arts are clearly remote from his own area of work. But perhaps he is entitled to take up the problem of beauty. For the adjective 'beautiful' is here indeed used to characterize the arts, but the realm of beauty reaches certainly far beyond its sphere of action. It surely encompasses also other areas of spiritual life – and the beauty of nature finds itself reflected also in the beauty of natural science. 
Perhaps it is good if we at first – without any attempt at a philosophical analysis of the concept 'beauty' – simply inquire where, in the area of the exact sciences, we can meet with beauty itself. Here I may perhaps begin with a personal experience. When I, as a young boy, was attending the lowest classes of the Max-Gymnasium in Munich, I took a great interest in numbers. It gave me pleasure to know their properties – for instance, to find out whether they were prime numbers or not, and to see whether they could be represented as sums of squares, or finally to prove that there had to be infinitely many prime numbers. Since my father found my knowledge of Latin much more important than my interest in numbers, he once brought to me from the State Library a treatise written in Latin by the mathematician Kronecker. In it the properties of numbers were put in relation to the geometric problem which consists in dividing a circle into a number of equal parts. I do not know how my father hit upon such a research from the middle of the past century. But the study of Kronecker's treatise made a profound impression on me. For I perceived it immediately as a thing of beauty that one could, from the problem of the division of the circle – whose simplest cases were already known to us from the classroom – come to learn something about the completely different questions of the elementary theory of numbers. In the distant background the question even presented itself briefly whether there are integers and geometric forms – that is, whether they exist outside the spirit of man, or whether they are just constructed by this spirit as instruments for the understanding of the world. 
But at that time I was not yet in a position to think of such problems. Only the impression of something very beautiful was quite direct, did not need any foundation or explanation. 
But what was beautiful here? Already in antiquity there were two definitions of beauty which stood in a certain opposition to each other. The controversy between these two definitions played a great role especially in the Renaissance. One of them designates beauty as the right concordance of the parts with each other and with the whole. The other, which goes back to Plotinus, without any reference to parts designates beauty as the translucence of the eternal splendor of the 'One' through the material phenomenon. In connection with the mathematical example we will have at first to side with the first definition. The parts are here the properties of the integers and the laws of geometric constructions. The whole is obviously the mathematical system of axioms which stands behind them, to which both arithmetic and Euclidean geometry belong. Namely, it is the great interconnection which is guaranteed by the freedom from contradiction of the axiomatic system. We realize that the single parts fit together, that they belong to this whole really as parts – and we, without any reflection, perceive as beautiful the compactness and simplicity of this axiomatic system. Beauty has therefore something to do with the age-old problem of the 'One' and the 'Many' which – once in strict connection with the problem of 'Being' and 'Becoming'– stood at the center of the early Greek philosophy. 

Since also the roots of exact natural science lie precisely in this area, it will be good to sketch the thought currents of that early epoch in rough outline. At the start of Greek philosophy there is the question of the fundamental principle – of the 'One' – out of which the manifold multiplicity of the phenomena can be made understandable. Although it may sound strange to us, the well-known answer by Thales – "Water is the material principle of all things"– contains, as Nietzsche has pointed out, three basic philosophical challenges that became important in the subsequent development. They are, first, that man should seek such a unitary fundamental principle; second, that the answer could be given only rationally, that is, without reference to a myth; and finally, third, that the material aspect of the world had here a decisive role to play. Behind these challenges stands naturally the tacit admission that understanding can always just mean one thing: to become aware of interconnections, i.e. unitary features, characteristics of relatedness, in the multiplicity. 
If, however, there is such a unitary principle of all things, then is one unavoidably driven to the question – and this is the next step in this conceptual development – as to how change can be made understandable out of it. The difficulty is particularly to be seen in the famous paradox of Parmenides. Only being is, non-being is not. But, if only being is, then there can be nothing besides being-something, that is, that would dismember this being, that would bring about changes. Therefore being should be thought of as eternal, uniform, temporally and spatially unlimited. The changes that we experience, then, could only be an appearance. Greek thought could not stand still long because of this paradox. The eternal succession of phenomena was immediately given, and people had to explain it. In the attempt to overcome this difficulty, various directions were tried by various philosophers. One development led to the atomic doctrine of Democritos. We want to cast just a very quick look upon it. Besides being, also non-being can be there as possibility, namely as possibility for motion and form – that is, as empty space. Being is repeatable, and thus we come to the view of atoms in empty space – the view which later became so endlessly fruitful as a foundation of natural science. But of this development we are not going to speak here any further. Rather the other development will be more precisely discussed: the one which led to the ideas of Plato and which brings us directly to the problem of beauty. This development begins in the school of Pythagoras. In it must the idea have arisen that mathematics – mathematical orderliness – was the fundamental principle out of which the multiplicity of phenomena could be made understandable. About Pythagoras himself we know little. It seems that his school was somehow like a religious sect. What can be traced back to Pythagoras with certainty is only the doctrine of metempsychosis and the establishment of religious-ethical prescriptions and prohibitions. In this school, however, the occupation with music and mathematics played an important part – and this was the decisive element for future time. In this connection Pythagoras must have made the famous discovery that equally tense vibrating strings produce a harmonic sound if their lengths stand to each other in a simple rational numeric relation. Mathematical structure, namely rational numeric relation as source of harmony – that was certainly one of the most pregnant discoveries that were ever made in the history of mankind. The harmonic resonance of two strings produces a beautiful sound. The human ear perceives the dissonance that originates from the disquiet of oscillations as disturbing – but it perceives the quiet of harmony, the consonance, as beautiful. The mathematical relation was therefore also the source of beauty. Beauty is – so reads one of the ancient definitions – the right concordance of the parts with each other and with the whole. The parts are here the individual tunes, the whole is the harmonic sound. Mathematical relation can then join together into one whole two originally independent parts and thus bring forth something beautiful. It was this discovery which, in the doctrine of the Pythagoreans, produced a breakthrough toward entirely new forms of thought. It led to the view that the principle of all being was no longer seen as a sensible stuff – as water was for Thales – but as an ideal form. Thereby was a fundamental thought expressed which later constituted the foundation of all exact natural sciences. Aristotle reports in his Metaphysics about the Pythagoreans: "They busied themselves at first with mathematics, promoted it and, being reared in it, considered mathematical principles as the principles of everything that exists. And seeing in the numbers the properties and foundations of harmony ... they conceived the elements of numbers as the elements of all things, and the entire universe as harmony and number.'' The understanding of the manifold multiplicity of phenomena, then, should come about through our recognizing, in it, unitary formal principles which can be expressed in the language of mathematics. Thus also a strict interconnection is established between intelligibility and beauty. For, if beauty is known as the concordance of the parts with each other and with the whole and if, on the other hand, all understanding can first come about through this formal interconnection – then is the experience of beauty almost identical with the experience of understanding or, at least, guessing such an interconnection. 

The next step in this development was undertaken, as is well known, by Plato through the formulation of his doctrine of ideas. Plato contrasts the perfect mathematical forms with the imperfect entities of the sensible world – for instance, the imperfectly circular orbits of the stars with the perfect, mathematically defined circle. Material things are the imitations, the shadows of ideal real structures. 
And thus – we would be tempted to continue – these ideal structures are real because and insofar as they are realizing in material events. Plato, then, distinguishes here with complete clarity a bodily being which is accessible to the senses from a purely ideal being which cannot be grasped through the senses but only through spiritual acts. However, this ideal being stands by no means in need of human thinking, so as to be brought about by it. It is, on the contrary, the authentic being after which both the corporeal world and the human thinking are patterned. The grasping of the ideas by the human spirit is – as already their name says – more of an artistic contemplation, a half conscious guessing than a rational knowing. It is the reminiscence of forms which have engrafted into this soul already before its earthly existence. The central idea is that of beauty and goodness, in which the divinity becomes visible, and at whose sight the wings of the soul grow. In one passage of the Phaedros is the thought expressed: “The soul is frightened, it shudders at the sight of beauty for it feels that something is conjured up in it which has not come to it from the outside through the senses, but has always been present in it, in a profoundly unconscious area”. 

But let us return again to the understanding and therefore to natural science. The manifold multiplicity of phenomena can be understood – say Plato and Pythagoras – because and insofar as here there are underlying unitary formal principles which are amenable to a mathematical representation. With that, the entire program of modern exact natural science is already anticipated. But it could not be carried out in antiquity because the empirical knowledge of details in natural events was largely lacking. The first attempt to come to grips with such details was undertaken, as is well known, by the philosophy of Aristotle. But, given the boundless fulness which at first sight offered itself to the observer of nature, and given the total lack of any viewpoints which could have made orderliness recognizable, the unitary formal principles which had been sought by Pythagoras and Plato had now to step back before the description of details. Thus already at that time comes to the fore the opposition which has lasted until now, for instance in the discussion between experimental and theoretical physics. It is the opposition between the experimentalist who, through accurate and conscientious spade work, creates the presuppositions for the understanding of nature and the theoretician who devises mathematical schemes by which he tries to order nature and thus to understand it. These mathematical schemes prove themselves to be the true ideas which underlie natural events – and this not only because they represent correctly the data of experience but above all because of their simplicity and beauty. Already Aristotle, as an experimentalist spoke critically of the Pythagoreans who, as he put it: “did not search for explanations and theories on the basis of facts, but rather, on the basis of certain theories and pet opinions, strained the facts and – so to speak – played themselves up as co-ordinators of the universe." Looking back at the history of exact natural science one can perhaps establish that the correct representation of natural phenomena developed precisely out of the tension between these two opposite conceptions. Pure mathematical speculation is sterile because it cannot find its way back from the fulness of possible forms to the very few forms according to which nature is really constructed. And pure experimentation is sterile because finally it is smothered by endless tabulations without intrinsic interconnection. Only from the tension that arises out of the interplay between the fulness of facts and the possibly suitable mathematical forms can decisive advances come. 

But this tension could not be taken up in antiquity and therefore the way to knowledge became for long time separated from the way to beauty. The meaning of beauty for the understanding of nature became clearly visible again only when people, at the beginning of modern times, succeeded in reverting from Aristotle to Plato. Only through this turn did the entire fruitfulness of the mental attitude introduced by Pythagoras and Plato reveal itself. Already the famous investigations of the fall – which Galilei did probably not carry out at the Leaning Tower of Pisa – indicate that most clearly. 
Galilei begins with accurate observations without taking into account the authority of Aristotle. Rather he tries, following the doctrines of Pythagoras and Plato, to find mathematical forms that correspond to the empirically obtained facts, and thus arrives at his laws of the fall. But in order to recognize the beauty of mathematical forms in the phenomena, he must – and this is a decisive point – idealize the facts or, as Aristotle had reproachfully put it, strain them. 
Aristotle had taught that all moving bodies come finally to rest if there is no action by external forces – and that was the common experience. Galilei asserts on the contrary that bodies without external forces persevere in the state of uniform motion. Galilei could dare to strain facts this way because he could point out that moving bodies are always exposed to the resistance of friction – and so motion lasts in actual fact the longer, the better the friction forces can be eliminated. Through this straining of facts – this idealization – he gained a simple mathematical law: and this was beginning of the modern exact natural science. 

A few years later Kepler succeeded in discovering – in the results of Tycho Brahe's very precise observations of planetary orbits – new mathematical forms. Thus he formulated his famous three Keplerian laws. How close Kepler felt in these discoveries to the old conceptions of Pythagoras, and how much the beauty of the interconnections led him to their formulation – this comes to the fore already in the fact that he compared the revolutions of planets around the sun to the vibrations of a string, and spoke of a harmonious resonance of the various planetary orbits: the harmony of the spheres. Finally, at the end of his work on the harmony of the world he burst into the cry of joy: "I thank you, God our Creator, for you have made me gaze upon the beauty of your creative work." Kepler was intimately seized by the fact that he had here come across a quite central interconnection: one not thougt out by man, and one whose first cognition had been reserved for him – an interconnection of highest beauty. 

A few decades later Isaac Newton in England brought this interconnection completely to light and described it in detail in his great work Philosophiae Naturalis Principia Mathematica. With that was the path of exact natural science marked out for about two centuries. But, is it not knowledge alone that is in question here – or also beauty? And if beauty, too, is in question what role has it played in the discovery of interconnections? Let us recall again the ancient definition: "Beauty is the right concordance of the parts with each other and with the whole." That this criterion applies in the highest measure to a construction like Newtonian mechanics, this hardly needs to be explained. 
The parts – they are the individual mechanical processes: those which we isolate accurately by means of apparatus just as those which unfold inextricably before us in the manifold play of phenomena. And the whole is precisely the unitary formal principle into which all of these processes fit the one which has been mathematically laid down by Newton in a simple system of axioms. Unitariness and simplicity are, to be sure, not exactly the same. But the fact that in such a theory the many are placed over against the one, that in the one the many are unified – this leads by itself to the consequence that the theory is simultaneously perceived by us as simple and beautiful. The meaning of beauty for the finding out of truth has been recognized and stressed at all times. The Latin motto, "simplex sigillum veri", "Simplicity is the seal of truth" stands in large letters in the physics auditorium of the University of Gottingen as a warning for those who want to discover something new. And the other Latin motto "pulchritudo splendor veritatis", "beauty is the splendor of truth" can also be so interpreted that the researcher first knows truth from his splendor shining forth.
Still twice in the history of exact natural science has this shining-up of the great interconnection become the decisive signal for significant progress. I am thinking here of two events in the physics of our century: the rise of the theory of relativity and that of the quantum theory. In both cases, after yearlong unsuccessful striving for understanding, a bewildering abundance of details was almost suddenly ordered. This took place when an interconnection emerged which, thought largely unvisualizable, was finally simple in its substance. It convinced through its compactness and abstract beauty – it convinced all those who can understand and speak such an abstract language. But we do not want to follow any further the evolution of history. Rather we would like to ask quite directly: What shines up here? How does it happen that in this shining-up of beauty in exact natural science the great interconnection becomes cognoscible –even before it can be understood in detail, before it can be demonstrated rationally? What does the shining force consist in and what does it accomplish in the further course of science? 

Perhaps at this point one should begin by recalling a phenomenon which can be called the unfolding of abstract structures. It can be clarified through the example of the theory of numbers which has already been mentioned at the beginning, but one can also point to similar processes in the development of art. For the mathematical foundation of arithmetic – the theory of numbers – a few simple axioms suffice which, properly speaking, just define exactly what counting means. But, with these few axioms, is in fact already laid down the entire fulness of forms that only in the course of a long history have entered the consciousness of mathematicians: the doctrine of prime numbers, that of square rests, that of the congruence of numbers, etc. One can say that those abstract structures, which are laid down by counting, have unfolded visibly only in the course of the history of mathematics – that is, history has brought to the fore the fulness of theorems and interconnections which make up the contents of that complicated science which is the theory of numbers. In a similar way, also at the beginning of an art style, for instance in architecture, certain simple basic forms are found, e.g. the semicircle and the square in the Romanesque architecture. Out of these basic forms emerge – in the course of history – new, more complicated, even changed forms which, however, can in some way be conceived as variations on the same theme. Thus, from such basic structures, a new manner, a new style of construction is developed. One has the feeling that the possibilities of development could be detected in these primordial forms already at the beginning. Otherwise it would be hardly understandable that many gifted artists decided quite rapidly to pursue these new possibilities.

A similar unfolding of abstract basic structures has doubtlessly taken place also in the cases I have enumerated concerning the history of the exact natural sciences. This growth, the development of continually new branches has lasted, in the case of Newtonian mechanics, until the middle of the past century. In our century we have experienced something similar in the theory of relativity and in the quantum theory – and the growth is not yet completed.
Incidentally, this process has also – in science as well as in art – an important social and ethical aspect. For many persons can actively participate in it. When in the Middle Ages a large cathedral had to be built, many architects and craftsmen were employed. They were filled with the perspective of the beauty which was laid down in the primordial forms, and were obliged, by their very task, to perform precisely accurate work in the spirit of such forms. 
In a similar way, during the two centuries that followed Newton's discovery many mathematicians, physicists and technicians had the task of treating individual mechanical problems according to the Newtonian methods, or of carrying out experiments or of undertaking technical applications. And here, too, always the greatest accuracy was demanded in order to attain whatever was possible within the framework of Newtonian mechanics. Perhaps one may say in general that through the underlying structures – in this case, Newtonian mechanics – guidelines are drawn or even value standards set according to which it can be objectively determined whether a given task has been performed rightly or wrongly. Just because of the fact that here precise requirements can be established, that the individual can co-operate – through little contributions – to the attainment of great goals, that an objective decision can be made about the value of his contribution: from all of this the satisfaction arises which actually flows, from such a development, to the large circle of the persons involved. Consequently, one should also not underrate the ethical significance of technology for the present time. 

Out of the development of natural science and technology has come forth, for instance, also the idea of the airplane. The individual technician who constructs any partial gear for the airplane – and the worker who produces it – knows that everything hinges on the extreme precision and accuracy of his work, that perhaps even the lives of many persons depend on its reliability. Consequently, he gains the pride which a job well done warrants. And he rejoices with us at the beauty of the airplane when he perceives that, in it, the technological goal has been implemented with the rightly suitable means. Beauty is – so runs the already repeatedly quoted ancient definition – the right concordance of the parts with each other and with the whole. But this requirement must also be fulfilled in a good airplane. But, by this reference to the development of a beautiful basic structure – and to the ethical values and requirements which later emerge in the historical course of such a development – the question posed before is still not yet answered. What shines up in such structures so that the great interconnection can be known, even before it is rationally understood in detail? By the way, the possibility should be excluded in advance that such a knowing may be subject to illusions. But that there is this immediate knowing, this fright in front of beauty – as Plato put in the Phaedros – it cannot be doubted at all. Among all those who have had investigated this question there seems to be a consensus that such an immediate knowing does not take place through discursive, that is, rational thinking. I would like to cite here at some length two statements. One is by Johannes Kepler of whom we have spoken before. The other, from our own time, is due to the atomic physicist of Zurich, Wolfgang Pauli, who was a friend of the psychologist C. G. Jung. The first text is to be found in Kepler's work Cosmic Harmony and reads: "That power which notices and knows the noble proportions in sensible objects and in other external things has to be ascribed to the lower region of the soul. It is very close to the power which provides the senses with their formal schemes or, even more deeply, to the purely vital power of the soul. It does not think in discursive way, that is deductively – as philosophers do – and does not employ any lofty method. Therefore it is not proper to man but also inheres in the wild beasts and domestic cattle ... One could then ask from what source that soul power which has no share in conceptual thinking and therefore also no proper cognition of harmonic relations may have the ability to know what is there, in the external world. For to know means to compare that which can be outwardly noticed through the senses with the inward primordial images and thus judge its concordance with them. Proclos has, in this connection, a very beautiful expression by using the image of waking up from a dream. The sensible things of the external world bring back to our memory those things which we have perceived before in the dream. In the same way the mathematical relationships which are present in sensible reality conjure up those intelligible primordial images which are inwardly present in advance. As a result they now shine up in the soul, efficaciously and vivaciously, while before they were present only in a nebulous way. But how have they penetrated to the interior of man? To this I answer – Kepler goes on to say – that all pure ideas or primordial formal relationships concerning the harmony of which we have been speaking inhabit interiorly those who have a capacity to apprehend them. But they are not first taken into the interior of man through a conceptual process. Rather they stem from one, as it were, instinctive and pure contemplation of sublimity. They are innate in these individuals, just as the morphological principle of plants – for instance, concerning the number of petals or the number of seed receptacles for the apple – is innate." 

Kepler, then, indicates here possibilities that may be already present in the animal and vegetable realms, namely inborn primordial images which bring about the cognition of forms. In our time Portmann, in particular, has discussed such possibilities. He describes for instance certain colorful patterns which are embodied in the plumage of birds and which can have a biological sense only if they are noticed by other birds of the same kind. The ability to notice, therefore, must be inborn just as the pattern itself. 
Here one can also think of the singing of birds. At first what is biologically required here is only a definite acoustic signal which serves, for instance, in the search for a mate and is understood by the mate itself. But in the measure that the immediate biological function decreases in importance, it may happen that the treasure of forms – namely, the underlying melodic structure – can acquire a playful enlargement and development. This then, as song, enraptures also such a biologically unrelated being as man. The ability to recognize the play of such forms must, in any case, be inborn in the bird species involved – it certainly stands in no need of discursive rational thinking. As regards man, to cite another example, he has probably the ability inborn to understand certain basic forms of gesticulation and to decide accordingly whether his neighbor harbors friendly or hostile intentions – an ability which is of the greatest significance for the social life of people. 

Ideas similar to those of Kepler are expressed in an essay by Wolfgang Pauli. Pauli writes: "The process of understanding in nature – as well as the delight that man feels in the act of understanding, that is, when he becomes aware of a new cognition – seems therefore to consist in a correspondence, a coming-to-cover-each-other: pre-existing internal images of the human psyche with external objects and their behavior. This conception of the knowledge of nature goes back to Plato, as is well known, and ... is also advocated very clearly by Kepler. He speaks in fact of ideas which pre-exist in the spirit of God and are created with the soul because of its being the image of God. These primordial images which the soul can perceive with the help of an inborn instinct Kepler calls archetypal. 
The agreement with the primordial images or archetypes which C. G. Jung has introduced into modern psychology as instincts of the active imagination is quite far-reaching. Modern psychology gives the proof that every understanding is a lengthy process which is attended by unconscious processes long before the consciousness content is capable of rational expression. By so doing, psychology has again directed the attention to the preconscious archaic stage of knowledge. At this stage, instead of clear concepts, images are present with a strong emotional content. They are not thought but so to speak, contemplated pictorially. The archetypes function as ordering operators and shapers of this world of symbolic images. They act as the sought-for bridge between sense perceptions and ideas and therefore are also a necessary prerequisite for the emergence of a scientific theory. However, one must beware of transposing this a priori of knowledge into the field of consciousness and relate it to definite ideas which can be formulated rationally." 

In addition Pauli discusses in the further course of his researches, the fact the Kepler came to the conviction about the rightness of the Copernican system not primarily because of detailed results of astronomical observations. Rather, what moved him was the correspondence of the Copernican image with an archetype – which C. G. Jung calls Mandala – that Kepler employs also as a symbol of the holy Trinity. 
God stands in the center of a sphere as the First Mover. The world, in which the Son is at work, is compared to the surface of the sphere. The Holy Spirit corresponds to the rays that run from the center to the surface of the sphere. Naturally, it belongs to the essence of such primordial images that they cannot be described in a precisely rational way. Even though Kepler may have attained his conviction about the rightness of the Copernican system also from such primordial images, a decisive prerequisite for any reliable scientific theory remains that it subsequently stand the test of empirical checking and rational analysis. On this point Kepler himself was entirely agreed. Here are the natural sciences in a happier position than the arts because for science there is an inescapable and inexorable criterion of value from which no piece of research can subtract itself. The Copernican system, the Keplerian laws and Newton's mechanics have subsequently proved their worth – through the interpretation of experiences, the data of observation and technology – with such an extension and such an extreme precision that their rightness could no longer be doubted ever since the appearance of Newton's Principia. Still, what is in question here is an idealization – of the kind that Plato had considered necessary and Aristotle had reproved. 

This fact came to the fore with complete clarity only about 50 years ago. Then people realized, through the experiences of atomic physics, the Newton's conceptual construction no longer sufficed to attain to the mechanical phenomena in the interior of the atom. Since the Planckian discovery of the quantum of action in 1900, a situation of bewilderment had arisen in physics. The old rules according to which man had successfully described nature for over two centuries resisted an adaptation to the new experiences. But also these experiences themselves were internally contradictory. A hypothesis which proved valuable in one experiment failed in another. The beauty and compactness of the old physics appeared destroyed. It seemed that, out of the often diverging attempts, it was not possible to obtain a genuine insight into new and different interconnections. I do not know whether it is permissible to compare the situation of physics in those 25 years after the Planckian discovery – which I still experienced as a young student – to the conditions of contemporary modern art. But I must confess that such a comparison forces itself upon me, time and again. The helplessness before the question what one should do with bewildering phenomena, the affiction over the loss of interconnections which still appeared so convincing – all this dissatisfaction has certainly determined the traits of the two so different realms and epochs in a similar way. However, what is in question here is obviously a necessary intermediate stage which cannot be skipped and which prepares a future development. For – as Pauli put it – every understanding is a lengthy process which is introduced by unconscious processes, long before the consciousness content is capable of rational formulation. Archetypes function as the sought-for bridge between sense perceptions and ideas. In the moment, however, when the right ideas emerge, a completely undescribable process of highest intensity comes to pass in the soul of the person who sees them. It is the astonished fright of which Plato speaks in the Phaedros. The soul, as it were, remembers what it had always possessed unconsciously. Kepler says: "geometria est archetypus pulchritudinis mundi". "Mathematics – we can generalize in translation – is the primordial image of the beauty of the world.'' In atomic physics this occurrence has been experienced less than fifty years ago. And it has restored exact natural science to a state of harmonious compactness under entirely new presuppositions – a state that had been lost for a quarter of a century. I do not see any reason why something similar should not happen one day in art. But one must add the warning: something like that man cannot make, it must happen by itself. 

Ladies and Gentlemen, I have described to you this aspect of exact natural science because in it the kinship to the fine arts becomes most clearly visible and because here the misunderstanding can be forstalled which sees science and technology as concerned only with exact observation and rational, discursive thinking. Of course rational thinking and accurate measuring do belong to the work of the investigator of nature – just as hammer and chisel belong to the work of the sculptor. But they are in both cases just tools, not content of the work itself. Perhaps at the close I may still recall once the second definition of the concept ‘beauty’ which stems from Plotinus and in which there is no mention of parts and whole. "Beauty is the translucence of the eternal splendor of the 'One' through the material phenomenon." There are important epochs in the history of art to which this definition applies better than to the aforementioned one. Frequently we feel a nostalgia for such epochs. But in our time it is difficult to speak of this aspect of beauty – and perhaps it is a good norm to adapt oneself to the customs of the time in which one has to live, and keep silent about that which is hard to say. In fact, the two definitions are not too remote from each other. Let us therefore be satisfied with the first more prosaic definition of beauty which certainly is verified also in natural science. And let us realize that beauty is – in exact natural science just as in the arts – the most important source of illumination and clarity. 

from a Lecture delivered to the Bayerische Akademie der Schönen Künste, München, July 9, 1970. Published as private paper in Stuttgart, 1971. Translation from German into English by Enrico Cantore.