Beauty

The notion of "beauty" (Lat. pulchrum; Gr. kalós ) in its various connotations can be found across the entire history of western thought, as a concept tied to aesthetic, logical, ethical and religious values. Beauty is defined as that which pleases the senses, the useful or the one matching the goal, the good, the true, the finding of an idea or its appearing, the divine and its epiphany, but also the harmonious, the proportionate, the one in the multiplicity. The idea of the affinity of beauty with the good can be found in Socrates' thought, especially in Plato's Symposium, it is emphasized and explained in Aristotle's Metaphysics (cf. XIII, 3), and then reiterated by Plotinus. The idea of beauty surfaces in Shaftesbury's ethics and aesthetic, and will influence the Romantic philosophy in its whole, due to the thought of Schiller. The Platonic notion of beauty in Phaedrus and in the Symposium triggers the renaissance discussions about love, it crosses Winckelmann's classicism, Schelling's metaphysics and Schopenhauer's philosophy. The idea of beauty as harmony, proportion, unity in multiplicity, probably Pythagorean in its origin, supported by Plato and Aristotle, can also be found in Thomas Aquinas' aesthetics. Yet, Aquinas doesn't devote too much attention to the notion of pulchrum . The good is sought after by willingness and it plays the role of a final causality. It is in beauty, which has instead the role of a formal cause, that the sight and the intellect find their rest (cf. Summa Theologiae, I, q. 5, a. 4; I-II, q. 27, a. 1 ad 3^um). In beauty, aesthetically intended, the intellect perceives the elements of integrity, completeness and perfection, the proportion among parts, the consonance with the subject, and thus a particular clarity and intelligibility.

The metaphysical interpretation of beauty as proportion, harmony, unity in multiplicity, is also linked to the problem of theodicy, as it occurs in Augustine and in Leibniz. The aesthetic or poetic value of beauty dominates 17th century meditation and becomes the central theme in Kant's Critique of Judgment. Here, the beauty of nature, outside art itself, becomes an independent concept and finds its realization in the elements of reality, such as plants, flowers, animals, landscapes, but also people, mental or moral behavior.  Kant ties up the feeling of beauty to the one of sublime. This latter is seen as a tool to perceive the perfection and the sense of the world, beautiful or ugly things, and thus used as a key to access the mysterious perfection of the cosmos and of God. In this case, beauty in terms of harmony and cosmic proportion becomes one of the hermeneutic keys of both philosophy and science. Both of them wanting to find the simplicity of a beginning: philosophy does it through the metaphysical language of transcendence, while science does it through the language of mathematics and the interpretation of the laws of nature in terms of order, coherence and unification.

I. Simplicity and Elegance: a Short Historical Account

1. Introduction. Simplicity and elegance are notions commonly used in the daily language. There is an abundance of proverbs, maxims and phrases in which simplicity and elegance are implied. "Simplicity" as a virtue appears to be a leading idea in many different domains: human character and conduct, explanation, style in general and style in the arts, and theory. "Elegance" is often used in the same contexts, and particularly stresses the component of human choice. In combination simplicity and elegance are seen as guiding ideas helping to attain results in the most general sense, results which fit optimally in a given real situation. Already in this spontaneous use of the notions of simplicity and elegance a number of tensions may be observed: the notions are connected with human activity, the results of which are nevertheless evaluated according to their consistency with reality; simplicity and especially elegance are hallmarks of careful training and discipline, but on the other hand they change into their contraries when not appearing as "natural" characteristics. Simplicity and elegance are not easily attainable, but should not show any tediousness. The notions of simplicity and elegance in everyday language are often coupled with mastery, which links them to the arts and to science.

Since the very beginning in the history of philosophy and science, simplicity and to a lesser degree elegance, are taken as ideals and principles to live, up to contemporary scientific theory and practice. Throughout of the history of science, the meaning and the importance of these notions have had quite differing emphases, but underlying these a continuity may be observed. The demand for simplicity, revealing the quest for elegance and beauty, is connected with the domain of "explanation" in general (demanding a minimum of explanatory elements), and more particularly with the "reasoning" and the "demonstration" (demanding a minimum of axioms and of steps), with "method" and "procedure" (demanding a minimum expenditure in general), and finally with "theory" (demanding a minimum of hypothesis, causes or assumptions). The need for such "minimum" is determined by the demand for adequacy. The motivation or foundation of this demand for simplicity may stem from different sources: convention and pragmatic utility, the experience that simplicity may be of a heuristic value, aesthetic motivation, ontological commitment, metaphysical or theological suppositions.

2. Aristotle and the Early Philosophical Thought. As the case with many other ideas and notions which play a leading role in philosophy and science, perhaps not so much the "birth" but also the maturation of the notion of simplicity can be found in the works of Aristotle (384-322 B.C.). From an historical perspective, "Aristotle's razor" is a fitter label than "Ockham's razor." Aristotle is also responsible for the fact that simplicity is a many-sided, complex or even muddled notion. His abundant use of some forms of the principle of parsimony, economy and simplicity is to be found in many domains and has many characteristics: as a principle of minimal ontological assumption (in his Physics), as a rule of method (cf. Posterior Analytics), as a criterion for theory evaluation (cf. De Coelo and Physics ), as a heuristic device (in the biological writings), as a surprising feature of the workings of nature, which gives aesthetic satisfaction and intellectual joy (the biological works). Moreover, he connects the demand for simplicity in its various forms with the rationality and intelligibility of nature. In a page of the Metaphysics, Aristotle replies to some philosophers who claimed that mathematics could not speak of the good and beauty. If mathematics, apparently, does not speak of beauty, he says, in reality it takes into consideration those same supreme conditions that constitute the forms of beauty: "[...] those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or their definitions, it is not true to say that they tell us nothing about them. The chief forms of beauty are order, symmetry and definiteness, which the mathematical sciences demonstrate in a special degree. And since these [e.g. order and definiteness] are obviously causes of many things, evidently these sciences must treat this sort of causative principle also [i.e. the beautiful] as in some sense a cause" (XIII, 3).

From the perspective of his personal theory of knowledge, simplicity as an ontological and epistemological characteristic might be qualified as a principle rather than an axiom or a rule. It was not seen as a simple objective, but as the culmination of a challenging intellectual attempt to search for whatever stands at the beginning. In this search for the principles, intuition plays an important role, drawn as it is by the intellectual satisfaction that comes with the progress made in the right direction. In this process, simplicity acts as an heuristic catalyst. Most of the concepts of the ongoing epistemological debate on simplicity can be connected with relevant texts in the corpus of  Aristotle works. These are, for instance, simplicity as a criterion in both the context of discovery and the context of justification (to use a modern jargon), as an indicator of beauty and of truth, as having heuristic, methodical, ontological, metaphysical and religious aspects. These texts also make aware of the fact that the notion of the world's rationality, of cosmic order, of finality and of economy are linked in a circular metaphysics.

Yet, Aristotle was certainly not the only representative of ancient thought who advocated simplicity. He himself considers simplicity as being a standing principle in mathematics. Also in Greek astronomy many instances can be found in which simplicity and economy play an important role. The study of the heavens in this science is linked to the Divine and to the characteristics of the Divine: simplicity and perfection. The dominant role of the circle and the sphere in Greek astronomy may be connected with these characteristics. The concept of the beautifully ordered cosmos, the importance of number and the beauty of astronomic method reflect divine perfection and simplicity. This belief has seasoned the scientific enterprise ever since. The application of the criterion of simplicity in astronomy also leads to tensions which have played a continuing role as well: the tension between simplicity of principles (both qualitative and quantitative) and basic concepts on the one hand, and the commodity and simplicity of operations and calculations on the other; the tension between truthful acknowledgement of facts, which endanger simplicity, and the self-destructive seduction to introduce auxiliary hypotheses in order to "save" simplicity.

A term containing positive and negative connotations in early Latinity and connected with choice in general, elegantia became during the times of Cicero (106-43 B.C.), a moral and intellectual virtue. Cicero's vir elegans ought to be a master in the art of choosing. Tasteful choice could and should be a natural characteristic (however much a product of training and discipline) of human conduct in many domains: language, rhetoric, art, friendship, morality, and science. Good taste in this sense also implies good and prudent judgment, it is not subjective or irrational, but instead it is the result of a moral and intellectual competence. This background is still present when we speak of an "elegant demonstration," an "elegant experiment" or an "elegant style." Elegantia in the Ciceronian sense is opposed to elegance as a shallow, artificial and studied appearance. It is connected with what is fitting and appropriate, without being superfluous (a demand which elegantia connects with the principle of parsimony) and without inadequate simplification. The notion of elegantia is always used in the context of human conduct and activity and the products thereof (also books and theories), but only in so far as they adequately fit reality.

3. The Philosophy of the Middle Ages and "Ockham's Razor." Robert Grosseteste (1175-1253) has been one of the pivotal figures between the incomplete inheritance of ancient philosophical concepts and scholasticism. As an avid reader, translator and commentator of Aristotle, as well as being eclectically interested and versed in the study of natural phenomena, he links the notion of economy of thought, method and nature on the one hand with those of beauty, light and Creation on the other. Grosseteste might be regarded the founding father of the Oxford Franciscan school in scholasticism, with  Roger Bacon, John Duns Scotus, Ockham and Chatton, as his (very different) heirs. Especially William Ockham (1280-1349), but also other representatives of the scholastic period paid an increasing attention to the principle of parsimony. This attitude was however accompanied by a change of emphasis; apart from stronger accents stemming from theology, parsimony was especially connected with explanations and the ontological claims present in demonstration and explanation. "Reification" of concepts, as well as the introduction of the parva res as real causes should be discarded as the introduction of superfluous elements. This change of emphasis did not fully overshadow the aesthetic and metaphysical sides of the notion of simplicity, especially in relation to God and to Nature as God's Creation. It would be wrong to assume that Ockham's interpretation of the principle of simplicity could be labelled as anti-metaphysical. The debate between Ockham and Chatton on the principle of parsimony is a prelude to aspects of the contemporary debate. The term "razor," connected with the principle of parsimony probably dates from the Renaissance, while the Latin formula entia non sunt multiplicanda sine necessitate is not found in Ockham's writings. Yet it is possible to demonstrate a continuity between Ockham's attention to the notions of simplicity and parsimony and its successive formulations and reinterpretations up to the 20th century.

4. The Birth and Development of Modern Science. This continuity of the use of simplicity and elegance as guiding-ideas will appear later on also in modern science. However, the rise of modern science is often interpreted as a revolt against aristotelianism and scholasticism. But, as it will become obvious in the next section (see below, II.2), in the recommendations of the new concepts and methods of the "revolutionaries," one may discern a continuity with the preceding period, in so far as parsimony and simplicity are concerned, often expressed in the very same maxims and phrases encountered in the Scholastic period, which drew them from Greek thought, especially from Aristotle. The recommendation concerning simplicity leads to several theoretical successes, in which a growth of simplicity and unification is accompanied by a growth of the domain of application and in fundamental insight. The central theories of Newton, Maxwell and Einstein illustrate this development. From the 18th century onward, the aesthetical aspect of simplicity in science, expressed by the demand for elegance in theory, method and experiment, was increasingly stressed. As a paradigmatic example of this attitude, Einstein's writings give an abundance of illustrations (see below, II.3). In the examination of this attitude it is striking that all elements present in Aristotle's implicit and explicit use of the principle of simplicity still play a role in modern scientific discourse: as a heuristic device, as minimal ontological commitment, as a criterion for theory-choice and theory-rejection, as an element of style and method, as a source of intellectual satisfaction, as an indicator of the world's rationality and intelligibility.

Gradually, the use of the notion of simplicity, and to a considerably lesser extent that of elegance, became a subject of reflection among scientists themselves and among philosophers of science. Especially the discussion in the latter category of scholars led to very diverging analyses and evaluations: simplicity as a myth, as a matter of convention, as an indicator of beauty, as an indicator of truth, as a principle of minimal assumption considered to be a principle of nature, as an expression nature's rationality. The disapproval of simplicity as a myth is falsified by the lively and fruitful use of this criterion, as convincingly demonstrated by the historiography of science. It proved difficult to characterize, systematize or even quantify the use of simplicity as a criterion in science. Apparently simplicity is a complex and basically a "relative" notion, and belongs rather to the domain of intuitive, common sense knowledge (Gr. phrónesis) than that of critical knowledge (Gr. epistéme). The discussion among philosophers of science did however lead to a measure of clarification: several forms of simplicity were distinguished, and within each form degrees of simplicity may be evaluated. In the reconstruction of decisions concerning theory-choice, it proves to be difficult to weigh these factors; especially because other factors, apparently of a non-rational type (aesthetic, metaphysical suppositions, etc.), play an additional and complex role. In recent years more attention has been given to the role and status of these aesthetic factors, the investigation of which has lead to a re-appraisal of so called non-logico-empirical factors, also called personal factors, in theory-choice.

II. The Quest for Beauty and Simplicity in Scientific Formulations

1. Aesthetical Experience and the Aesthetic of Formulas. The short historical itinerary traced here underlines how, since ancient times the work of study and of interpretation of nature has witnessed a particular convergence between simplicity and knowledge; in such convergence beauty, in its connections with elegance and harmony, plays a particular role. The birth of modern science did not imply any interruption in this respect. As A. Lamouche (1955) already showed in the past, the scientists who have honored the "principle of simplicity" - because they saw in it an heuristic principle of truth and comprehension of reality- are too many to be mentioned. A big volume would be necessary to report only the most meaningful quotations. From Copernicus to Einstein, scientists' reflections on nature and their seek for physico-mathematical formulations capable of representing the observed phenomena, shows the persisting presence of the following attributes: simplicity, elegance, harmony, order and unity, almost always related to a precise conviction concerning the rationality and the intelligibility of the world. The language and the modalities with which these attributes are used, surprisingly resemble those encountered in Aristotle's works or in the writings of Medieval and Scholastic authors.

The relationship between beauty and scientific thought is, in a certain way, wider and deeper than what mere scientific formulations might suggest. First of all, such a relationship stems form the very observation of nature, which is often the place of a true and unique aesthetic experience. Later on, the scientist becomes increasingly aware that a good deal of such beauty responds to symmetrical and harmonious criteria which can be expressed in terms of geometrical rules and mathematical formulas. This is what happened, for instance, in the field of music since Pythagoras onward. Here, the beauty of harmony was put in relation to specific mathematical relationships between frequency and amplitude of sounds. Working in tune with this perspective,  Kepler interpreted the planetary motions around the sun as a kind of "cosmic music," giving his treatise the meaningful title of Harmonice Mundi (1619). Beauty in mathematics and geometry may be recognized in some regular forms occurring in nature. So are Archimedes' spiral, the ratio amongst the different lengths of some parts of a living organism, which run according to the "gold section" ratio, or follows the "Fibonacci series," as occurs in the structures of many vertebrates and in the ramifications of not a few number of trees and other vegetal species. A privileged place of the dialogue between geometry and beauty is architecture and, in more recent years, engineering. Just as it happened for the cathedrals of the past, and now in the sophisticated structures of contemporary bridges and aeroplanes, beauty and harmony of aesthetics are often directly proportional to the "quantity of science" that they contain. To examples of the unwavering call for an aesthetic experience that comes from science, well known to the general public, is nowadays witnessed by the great diffusion of fractal geometry (frequently used also in many extra scientific domains) and by the great success achieved by astronomical images of nebulas and galaxies taken by the orbiting Hubble Space Telescope and then transmitted all over the world.

2. Some Examples of the Quest for Beauty in the History of Science. An aspect which deserves to be considered is the relationship that simplicity and beauty engage with the language of the sciences, especially when theoretical formulations are concerned. The relationship is so tight that it has lead Paul Dirac to claim that "it is more important to have beauty in one's equations than to have them fit an experiment. [...] It seems that if one is working from the point of view of getting beauty in one's equation, and if one has really a sound insight, one is on a sure line of progress" (Dirac, 1963, p. 47). Going back to the birth of modern sciences -presented by some authors as a "revolution" triggered by Copernicus, initiated by Kepler and by Galileo and finally put into effect by Newton- it must be said that all these "revolutionaries" substantially agreed with Aristotle. At least they did so when their conversation dealt with simplicity, order and beauty of the cosmos. In talking about his heliocentric system, Nicholas Copernicus wrote: "In the middle of all dwells the Sun. Who indeed in this most beautiful temple would place the torch in any other or better place than one whence it can illuminate the whole at the same time? [...] We find therefore, under this orderly arrangement, a wonderful symmetry in the universe, and a definite relation of harmony in the motion and magnitude of the orbs, of a kind it is not possible to obtain in any other way" (De Revolutionibus Orbium caelestium, I, cap. X). It is in referring to the Copernican system that Kepler is said "to have contemplated its enchanting beauty with a great joy" (cf. Opera Omnia, Frankfurt 1858, vol. VI, p. 116). Copernicus' proposal to put the sun at the center of his system was the answer to criteria of simplicity aimed at reducing the number of epicycles needed to reproduce the movement of the planets on the celestial sphere; just as Galileo did in his table-top experiments, while searching for a law that would regulate the increase of the speed of rolling bodies in the simplest way (cf. Opere, edited by A. Favaro, Firenze 1968, vol. VIII, p. 197). Likewise, Newton's universal gravitational law was intended to answer a great unification criterion: the linking between the attraction of falling bodies by the earth and the laws which govern the motions of celestial bodies. This procedure has been well documented in a number of occasions by Kepler: "nature loves simplicity," "it loves unity," "there is nothing that has to do with laziness nor waste in it," "to carry out its tasks nature always chooses the easiest manner" (cf. Opera Omnia, vol. I, pp. 112f). All these expressions can be easily found also in Aristotle's philosophy of nature.

In the modern age, one of the clearest examples of beauty and symmetry in the formulation of a scientific theory is probably that of the electromagnetic field equations, as they have been expressed by  J. Clerk Maxwell (1831-1879). His theory unifies various phenomena belonging to both electric and magnetic fields, whose nature would seem, apparently, independent. One of the equations which embodies at the best the high aesthetic value of the theory is that connecting the speed of light c with the dielectrical constant ε₀ and the diamagnetic constant μ₀ of the vacuum, according to the formula c ²   ε₀ μ₀ =1 , obtained on purely theoretical basis and confirmed later on by experimental measuring. In the realm of chemistry, from the mid 19th century, and above all with the beginning of atomic physics at the beginning of the 20th century, scientists were witnesses of how, in nature, it was taking shape what we know now as the "Periodic Table of Elements", and whose first development took place in 1870 through the work of D.I. Mendeleev and L. Meyer. The Table represents a unique example of simplicity, internal cohesion and explicative power. Based on the progressive order of the atomic weight and ranked by a unit which corresponds to the simplest element, the hydrogen, all the places of the Table are occupied by the various chemical elements. Along the rows of the Table, the chemical properties of the various elements are periodically reproduced, according to growing levels of complexity, perfectly related --as discovered later on-- to the structure of the electronic orbitals associated to the nucleus of each chemical element. We are facing a "unique" descriptive plot capable of unifying and connecting among each other the properties of the 92 chemical elements found in nature, from hydrogen to uranium, and that can be extended to those other elements we now synthesize in our labs. The discoveries of isotopes and of the fine structure of electronic orbitals have not modified the original pattern of the Table, having enriched it while maintaining its logic. This makes it possible to a 21st century chemistry student, although he has acquired much more knowledge, to identify and put in order such information in a much easier manner than it was possible to a chemistry scholar at the beginning of the 19th century.

Additional examples of elegance and beauty in the theories of physics are those related to the atomic theory of Bohr (1913), who was able to explain the complicated structure of the emission spectra of excited atoms employing an ordered sequence based on a precise group of "quantum numbers." As well, the theory of crystals, which predicted that the regularities observed on a macroscopic scale were to be explained by regularities found on the microscopic scale, at the level of ordained structure of atomic and molecular bindings forces. Criteria of symmetry also apply to analytical mechanics, where translation, rotation and similarity symmetries can be tied in an elegant and simple way with the invariant properties and the laws of conservation. Then we have the wide landscape of fractal description, based on laws of symmetry and replication rules. Contemporary quantum mechanics expresses most of its formulations within the "Group Theory", where the symmetry between properties of the various particles still plays a fundamental role and can successfully suggest the existence of new particles even before they have been experimentally discovered.

3. The Case of Albert Einstein. Within our subject matter, a very special and well-documented example is that of  Albert Einstein (1879-1955). One of his best known biographers, Abraham Pais, claims that, both the special and general theories of Relativity, as well as his constant search for a "unified field theory," originated from a well defined aesthetic concern: "Einstein was driven to the special theory of relativity mostly by aesthetic reasons, that is, arguments of simplicity. This same magnificent obsession would stay with him for the rest of his life. It was to lead him to his greatest achievement, general relativity, and to his noble failure, unified field theory" (Pais, 1982,  p. 140). Contemplating Einstein's remarks on beauty and simplicity, scattered over a period of 45 years of scientific activity, one can discern a number of aspects which are however closely related and are all based on a common ontological assumption. For Einstein simplicity was perhaps foremost a heuristic guide, both in terms of method and principles. According to him, good theories have "simple" origins.

He was particularly attracted by the high aesthetic value of Niels Bohr's atomic theory, used in physics from 1910 to 1920. He thought that the fact that Bohr skillfully and instinctively succeeded in discovering the principal laws of the spectral lines and of electronic orbitals, relying on the meaning and the explicative power his theory had for chemistry, was "like a miracle" and it represented "the highest form of musicality in that sphere of thought" (Pais, 1982, p. 416). Einstein had an analogous appreciation towards Planck's theory of thermal radiation, justified on the basis of its simplicity and its analogy to the classical theory. Looking for a complete unification between the electromagnetic and the gravitational fields, he was supported by the total confidence that the link "must" have been present in nature, because the experience carried out until then justified the intuition that the ideal of simplicity applied in nature. This assumption was basically of an "ontological" kind, one which would have certainly pleased both Plato and Aristotle. Simplicity seemed to guarantee a threefold function: as a signal of validity, as a heuristic and methodical tool, and as a road towards the unification of the laws. This was nothing but a new proposal in modern terms of the ancient rule simplex ratio veritatis, simplicity has the value of truth.

The ontological assumption of simplicity is accompanied, and somewhat caused, by an aesthetical impulse. Commenting on his work toward a unified field theory, Einstein stated that "its purpose was neither to incorporate the unexplained nor to resolve any paradox. It was purely a quest for harmony" (Pais, 1982, p. 23). Motivations of aesthetic and emotional nature have undoubtedly played an important role in the origin and development of Einsteinian theories. In his letters and speeches he repeatedly underlined the influence of these aspects, and he assumed that this motivation was overtly and secretly recognized by his colleagues. Addressing  Max Planck on the occasion his 60th birthday, Einstein said: "The longing to behold [...] preestablished harmony is the source of the inexhaustible persistence and patience with which we see Planck devoting himself to the most general problems of our science, without letting himself be deflected by goals which are more profitable and easier to achieve. I have often heard that colleagues would like to attribute this attitude to exceptional will-power and discipline; I believe entirely wrongly so. The emotional state which enables such achievements is similar to that of the religious person or the person in love; the daily pursuit does not originate from a design or program but from a direct need" (Pais, 1982, pp. 26-27).

Judged upon more general basis the attitudes towards simplicity, beauty and elegance showed by the founder of the theory of Relativity seem, without any doubt, well balanced. It is not about being obsessive, but rather a guiding-principle that experience has proven valid and that experience itself, in the future, could prove to be insufficient. With this respect, the "scientific faith" in the reality and the objectivity of nature seem to remain the main paradigm of the entire Einsteinian approach. Einstein, for example looked at one of his attempts towards unification as "a very beautiful theory, but filled with doubts" and he considered the first formulations of Bose's quantum statistics "an elegant deduction, but whose essence remains obscure." On December 1954 he labelled one of the last formulations of field equations as follows: "In my opinion, the theory presented here is the logically simplest relativistic field theory which is at all possible. But this does not mean that nature might not obey a more complex field theory" (Pais, 1982, p. 349). This is quite a prophetic affirmation when compared to contemporary theoretical efforts to reach to an adequate formalism for a Grand Unified Theory (GUT). A theory in which ever more complex, but also more general and elegant, multidimensional spaces are introduced, as displayed by the successful using of the 10-dimensional space in some contemporary versions of "superstrings theories."

III. Further Philosophical Reflections about Beauty and Simplicity

The quest for beauty represents one of the most important "bridges" to overcome the gap between scientific and humanistic disciplines that characterizes our contemporary culture, although some indication exists that such a gap is now shortening. Beauty is an issue that intrigues both great scientists and (most) artists. Van den Beukel's description of Maxwell's equations speaks of a "miracle of beauty, of conciseness and of concentrated expression" (De Dingen hebben hun geheim, Baarn 1990, p. 31). This description is equally apt for one of Shakespeare's sonnets or one of Haydn's string quartets. No doubt then, that the role of beauty is quite ample: a number of theories and principles have been chosen because of their aesthetic value not only in mathematics or natural science, but also in psychology, sociology or economy.

1. Objective and Subjective Dimensions of Beauty. Pondering about beauty comes endowed with strong personal components. One may safely state that aesthetical satisfaction has been one of the main roots of all kinds of speculative activity since Pythagoras and Plato, and that a considerable part of intellectual endeavor is reigned by "aesthetical passions" rather than by cool and distant rationality. This has on several occasions led to a blinding by these passions, an example of which would be the passionate belief concerning the perfect circularity of celestial spheres and planetary orbits, which clashed later on with the reality of the facts. It would also be unreasonable to try reducing all of quantum physics to one single pair of elementary particles. Along the same line, it would be inappropriate to nurture a misleading desire of reductionist simplicity when thinking that scarcity is the one and only explaining principle in economics, or Freud's libido the only and key-principle to understand all the activity of human psyche.

According to one school in philosophical psychology, beauty in creative art and beauty in the study of nature are governed by the same mental laws and relations. In this idealistic approach, beauty, symmetries, laws, and relations, are seen as contributed by human mental "machinery," and not as present in nature itself, if "nature itself" would at all be accessible without the mind's contribution. If such a contribution could be described by the general label of ordering complex data of whatever kind, one could consider art as merely an activity that "organizes" visible, audible and tangible phenomena, and science (in general) as the ordering of mental phenomena: thought. One could ask, however, why aesthetic canons are capable of guiding so directly the scientist towards the comprehension of reality. If "unifying and synthetical" intuitions have been successful in science it is because nature itself admits unification and is better understandable by means of symmetries. Although aesthetic canons may come with historical and cultural changes (what nowadays would be accepted as beautiful and elegant could be rejected in the future), there is a basic perception of the concept of beauty that survives over the epochs and changes, and which is common to all people who try to approach reality.

The relationship between subject and object, just as the one between truth and beauty, are not easy to decode. We can however affirm that aesthetic dimensions of science are indicators of both beauty and truth. In the beginning of a cognitive process, the subjective dimension is an important factor which, guided by intuition and its synthetic unifying capacity, brings together into a fitting unity what at first sight appears in nature to be dispersed and scattered, or recognizes recurrent models and forms (patterns) in that which would appear as random and chaotic. In this process intuition is stimulated by rhythm and harmony, in which harmony has a half-aesthetical, half-logical meaning, connected with our sensitivity for proportion and measure, and rhythm is experienced as (supposedly) law-regulated repetition. It is indeed striking how many natural processes can be described successfully in terms of "frequency." Rhythm in this sense is conducive to intellectual economy and simplicity: because of the element of repetition, which is implied by the notion of rhythm, many instances can be described correctly by an adequate knowledge of one cycle of the rhythm. Through rhythm, nature knows how to produce what is new out of the same, and what is complex out of the simple. In such a line of speculation, which Pythagoras, Grosseteste and Kepler would probably applaud, the universe may be seen as a cosmic musical composition, which science has to decompose (for understanding and explanation) and to recompose (for application), both being procedures in which simplicity and elegance are fruitful guides and criteria.

2. Aesthetic and Epistemic Simplicity . A further development comes from the distinction between "epistemic" and "aesthetic" simplicity. The former is supposed to be rational, in a sense measurable or at least comparable, and belongs to what has been labelled as the "context of justification of a theory." The latter form of simplicity cannot be formalized in pure rational terms, it is not quantifiable and it belongs to the "context of the discovery." Aesthetic simplicity is connected with the striving for intellectual and theoretical beauty; epistemic simplicity comprises a number of forms of notational, semiotic, and syntactic simplicity. Elliot Sober (1975) maintains that these two forms are really one, because they share a common logic structure. But aesthetic simplicity would then be irrelevant to epistemic simplicity and thus the former would end up being absorbed in the latter. Both Dirac and Einstein would strongly disagree with this conclusion. James McAllister (1989) suggests to classify simplicity as one of the "indicators of beauty," together with symmetry, analogy and principles of metaphysical kind. We would also have "indicators of truth," such as internal consistency, consistency with existing and well-corroborated predictive accuracy, pragmatic fruitfulness. The two kinds of indicators belong to different categories and they should not be readily mixed. While the first indicators would be subject to historical changes, the second ones would show a higher degree of stability. Indicators of beauty would be mainly chosen by "conservative" scientists.

The somewhat extreme positions maintained by the two authors mentioned above may be overcome. In fact, countless examples from the history of sciences bring about two critical observations. First of all, many scientists experience beauty, simplicity and tension towards truth as something intricately connected and interwoven. Secondly, aesthetic canons are not totally reducible to the epistemological level, as they play a leading role in causing not only scientific revolutions, but also the activity of science as such. In one of his subsequent works, McAllister has modified his analysis of the role influenced by the study of historical examples (cf. McAllister, 1990). The relationship between the objective and subjective aspects of beauty in the sciences gets summarized by Cantore (1977) who affirms that scientific beauty is not something abstract, but rather actual, because it is discovered in observed objects found in reality; nor is it merely intellectual, because it consists of models that can be perceived by the trained senses, at least indirectly. Scientific beauty is a beauty that can be consistently seized through intellectual research, but it can be also found at the basis of sensible knowledge, as beauty is commonly intended.

3. A Short Summary. Further philosophical reflections show that there exists an intricate relationship between the assumed role and status of simplicity and beauty, and the ontological suppositions of one's epistemology. The claim that the success of simplicity as a criterion in the contexts of investigation and justification can be founded on the innate order and simplicity of nature, implies a realist epistemology. In such a view, simplicity is a "bridge" between the conceptual and the real, and the sense of simplicity and beauty is a sign of the "resonance" between human rationality and the rationality of the ordered cosmos. Scientific endeavor tries to reach a harmony between these spheres, a quest in which simplicity is a guide. Under such a point of view, the sense for simplicity is an almost metaphysical intuition for what is fitting and right. From this perspective, an indicator of beauty is to be identified with an indicator of truth. Such a claim for the role of simplicity and beauty in science is not accepted by those who endorse a view strictly tied to positivism. Nevertheless, the historiography of science indicates that a majority of scientists implicitly or explicitly adhere to a realist epistemology, in which simplicity and beauty are more than mere pragmatic tools. The abundance of indications stemming from the history of science on this subject tells us that the appeal of simplicity and beauty is a result that must be expected, considering the roots and the "history of the effects"  (Ger. Wirkungsgeschichte) of the notion of simplicity.

The search for simplicity however, should be checked. A.N. Whitehead put it in a provocative way: "The aim of science is to seek the simplest explanations of complex facts. We are apt to fall in the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be: seek simplicity and distrust it" (The Concept of Nature [1920], [Cambridge: Cambridge University Press, 1955], p. 163). This attitude is a common intellectual temptation and a danger, which has devoured many a leading scholar, to go for the whole of reality from the perspective of one passionately supported principle (see above, n. 1). Simplicity should not be idolatrized: the notion contains anthropomorphism, and it implies selection, reduction, idealization, construction and often distortion and mutilation. The many-sided Razor should be used prudently, including good intellectual taste and judgment. The frequent combination of the notions of simplicity and elegance may be read as a signal that common sense should check simplicity. Elegantia , as the art of choosing and the expression of good judgment (Gr. phrónesis), is of an even higher rank than simplicity understood as a criterion in science (Gr. epistéme). The important sense of a competent and prudent choice keeps the scientist aware of what simplicity leaves out when used in operation. This awareness should be increasingly stressed the more complicated is what comes under the mind's attention.

IV. Beauty and the Divine

1. The Quest for Beauty as Part of Religious Experience. Many scientists have been able to make a step forward, passing from the acknowledgement of an aesthetic dimension in science, to the recognition that such a dimension seems to gain, at times, a "religious" nuance. If in the unity of the scientist's personal experience, scientific competence, aesthetic sensibility, and religious feelings coexist together, then it is hard to establish a clear cut separation between these domains. A concise expression of such interaction can be found in Lord Rayleigh's remark: "Some proofs command assent. Others woo and charm the intellect. They evoke delight and an overpowering desire to say 'Amen, Amen.'" (quoted by Huntley, 1970, p. 6). Aesthetic and existential-religious dimensions of nature are also linked in the following remark by Henri Poincaré: "The scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in such a study; and he takes pleasure in it because nature is beautiful. If nature were not beautiful, knowing about it would not be worth while and life would not be worth living" (quoted by Chandrasekhar, 1979, p. 25). The exercise of rationality, fascination and intellectual joy before nature, the desire to adore, stand for many authors as different, but contiguous degrees on the same ascending line. A beautiful theory and elegant solution (as well as a well-chosen experiment) is experienced by its intellectual creator as a marvel of nature, rather "given" than as the rational product of his or her own making. No doubt that such a blurring of demarcation lines and mixing of categories will be a sore in the eye for those who maintain a strong positivistic inclination.

Generally speaking, rational knowledge and religion are two separate things in the philosophic realm. The Dutch philosopher Frits Staal prefers to distinguish two main currents: the purely philosophical-critical approach and the religious approach. "I call this second current 'religious,'" he says, "because the main difference between philosophy and religion is that the former attaches a meaning or a sense to as few matter as possible and the latter to as many as possible" (Over zin en onzin in filosofie, religie en wetenschap, Amsterdam 1986, p. 12). The validity and the opportunity to distinguish these two fields does not get questioned in this case. What Staal considers to be a characteristic of pure philosophy, he a fortiori might attribute to science, but when we observe practice and mores in scientific enterprise, Staal's distinction is confirmed. In experiment (at least in its published description), in method, in mathematical notation and reasoning, in technological application, there is little or no room for ambiguity, emotion and aesthetical and religious evaluation. However, when scientific "motivations" are concerned, as well as the "appraisal" of science's results, method and foundation, it is not uncommon for scientists to include and bring in religious elements, putting them into relation to that wider and more general human enterprise which is the search for truth. According to Van den Beukel, "science, art and religion are all three occupied by the riddle of human existence, by his surrounding world, and by the relation between these two" (De Dingen hebben hun geheim, Baarn 1990, p. 131). Science is part of culture, like art and religion are. They cannot be considered as being strangers to or enemies of each other (there is ample historic evidence to contradict the famous gaps between these three "cultures"), instead they can enforce and stimulate each other. That this has often been the case is illustrated by the history of the criteria of simplicity and elegance. For this reason it is not surprising that also three supposedly incompatible "language-games" of these domains may intermingle: "Culture calls forth works of great beauty... Poetry that contains no superfluous word... Maxwell's laws, those jewels of mathematical beauty and eloquence... Simplicity gained from chaotic phenomena, in which Newton recognized the simplicity of his Creator. Wonders which make man wonder" (ibidem, p. 174).

A great supporter of the principle of simplicity, Isaac Newton increasingly stressed its importance in the three subsequent editions of his Principia. In the Opticks he did a further step by formulating quite fundamental questions. In this work, after blaming the "most recent philosophers" responsible for multiplying hypotheses, he claims: "The main business of natural philosophy is to argue from phenomena without feigning hypotheses and to deduce causes from effects, till we come to the very first cause, which certainly is not mechanical: [...] whence is it that nature does nothing in vain; and whence arises all that order and beauty which we see in the world?" (Opticks , 1730, Query 28). Newton asks whether these properties stem from the direct operation of God, and like Aristotle and (among others) Ockham did before him, he tries to find theological grounds for the postulate of the simplicity of nature. In this enterprise, Newton is not one of the last survivors of a dying genus of believing scientists, but rather - also in this respect - an influential and impressive hinge in the continuity of cultural endeavor, in which art, science and religion are experienced as interconnected parts. Although the image he had of God wasn't in agreement with the content of Christian Revelation, his view continues to intrigue scientists today. Addressing non-believer scientists, John Polkinghorne points out that the encounter one has with "rational beauty" through science, is similar to a religious experience, and among believers it has determined a coming back to natural theology, at least among physicists (cf. Spiritual Growth and the Scientific Quest, in "The Way," October 1992, p. 256).

A contemporary example of how the issue about beauty is still valid nowadays is provided by the physicist Henri Margenau. He emphasizes that beauty can hardly be justified in terms of chance or of struggle for survival, that is the two leading paradigms often used to explain the origin of the many shapes and varieties found in nature. "'Why is there so much beauty in nature?' We do not believe that beauty is only in the eye of the beholder. There are objective features underlying at least some experiences of beauty, such as the frequency ratios of the notes of a major chord, the symmetry of geometric forms, or the aesthetic appeal of juxtaposed complementary colors. None of these have survival value, but all are prevalent in nature in a measure hardly compatible with chance. We marvel at the songs of birds, the color scheme of flowers (do insects have a sense of aesthetics?), of birds' feathers, and at the incomparable beauty of a fallen maple leaf, its deep red coloring, its blue veins, and its golden edges. Are these qualities useful for survival when the leaf is about to decay?" (The Miracle of Existence [Woodbridge, CN: Ox Bow Press, 1984], pp. 29-30).

2. Beauty in Sacred Scripture and in Theology. Judaeo-Christian Revelation provides an answer to the question on the origin of beauty in a page of the  Book of Wisdom. The author blames those who were not able to recognize the Artificer of the world starting from the contemplation of created things, "but either fire, or wind, or the swift air, or the circuit of the stars, or the mighty water, or the luminaries of heaven, the governors of the world, they considered gods. Now, if out of joy in their beauty they thought them gods, let them know how far more excellent is the Lord than these; for the original source of beauty fashioned them. Or if they were struck by their might and energy, let them from these things realize how much more powerful is he who made them. For from the greatness and beauty of created things their original author, by analogy, is seen. [...] For they search busily among his works, but are distracted by what they see, because the things seen are fair" (Wis 13:2-5.7). Sacred Scripture contains a number of references to beauty, especially in the context of praising God in the works of creation, as found in the Psalms and in the "Wisdom Books". The beauty of the human being is also celebrated (cf. Gn 12:11; 39:6; 1Sm 16:12), as well as the beauty of human love (cf. Sg 1:14; 4:7; 7:6), the beauty of the divine law and of life in God's house (cf. Prv 3:17; 22:18; Nm 24:5), and the beauty of those works that human beings are able to carry out by virtue of their transcendent dignity. But above all it is worth mentioning the convergence between goodness and beauty, to the point that the two terms are at times almost interchangeable. At the end of each of the six days that categorize the narration of the creation (cf. Gn 1:9-31), the Jewish expression "God saw how good it was" — when talking about the creation of man and woman "he found it very good" — gets translated by the LXX Greek version as "God saw how beautiful it was (Gr. kalós)."  It is still close to the biblical text affirming that the heaven and the earth are made by God like "a work of art," a picture in which the human being occupies the highest point. When art or science seize the world in this same way, they are about to understand its characteristic of being a "creature," something capable of leading back to the mind and plan of an Artist. Yet beauty can, at times, mislead. Evil presents itself disguised as good in the surface; beauty, also, may remain confined into the sensible sphere, without going in depth: human beings stop at appearances, while God reads the hearts (cf. 1Sm 16:7). Beauty is not exempt of facing reality and has to come back to it always.

Christian theology has taken up the biblical message on aesthetics, merging it with Platonism and other sources of ancient philosophy, thus teaching the convergence between goodness, truth and beauty. Together with unity, those three attributes are nothing but the four classical "transcendentals of the Being," which theology ascribes par excellence to God, as cause and fullness of Being. All creatures, according to different degrees of perfection, participate in these attributes. They are, nevertheless, "attributes of God," that is, different ways human reason tries to approach God's nature, different perfections which, in Him, get identified into a one and only simplicity. In such a theological view there is no conflict, generally speaking, between truth and beauty. Beauty becomes a path to get to truth, and the knowledge of truth leads to the experience of joy and beauty. Truth is revealed by beauty and beauty is the fruit of truth. If beauty may distract from true goodness, it does so for an error in judgment, due to the limits and fallibility of the subject. We must finally remember how much Scholastic theology (indebted to both Platonic and Aristotelian approaches) exalted "God's simplicity." When the Being of God is understood (but not comprehended) as Pure Act, His highest "simplicity" gets emphasized. In God there is no composition between matter and form, because in Him there is no matter and He is form because of His essence. Nor is there any composition between essence and existence, because His being is His existing (cf. Summa Theologiae, I, q. 3, aa. 2 and 4). In Thomas Aquinas' theology, however, the good seems to have a priority with respect to beauty. The two concepts remain distinct. "Goodness properly relates to the appetite (goodness being what all things desire), and therefore it has the aspect of an end (the appetite being a kind of movement towards a thing). On the other hand, beauty relates to the cognitive faculty; for beautiful things are the ones that please when seen. Hence beauty consists in due proportion; for the senses delight in things duly proportioned [...]. Now since knowledge is by assimilation, and similarity relates to form, beauty properly belongs to the nature of a formal cause" (ibidem, q. 5, a. 4, ad 1^um).

Especially thanks to H.U. von Balthasar, during the last decades contemporary theology has undertaken a process aimed at reassessing the transcendental of pulchrum as a specific character of divine revelation as a whole, and as a privileged way to access to God (cf. The Glory of the Lord. A Theological Aesthetics, 1961-1965). Beauty is mainly related to Glory (Heb. kabod Jahvè), which creation portrays; this Glory also represents the primary appeal to knowledge about God as the epiphany of His intimate nature and as the ecstasy that ravishes human beings and invites them to participate in the divine life.

Finally, on the relationship between theology and beauty, John Paul II's Letter to Artists (1999) must be mentioned. In this letter, the Roman Pontiff develops the parallelism between goodness and beauty, between aesthetical and religious experience. In a page of this letter, which could have been addressed and applied to people of science as well, we can read: "Every genuine inspiration, however, contains some tremor of that 'breath' with which the Creator Spirit suffused the work of creation from the very beginning. Overseeing the mysterious laws governing the universe, the divine breath of the Creator Spirit stretches to human genius and stirs its creative power. He touches it with a kind of inner illumination which brings together the sense of the good and the beautiful, and he awakens energies of mind and heart which enable it to conceive an idea and give it form in a work of art. It is right then to speak, even if only by analogy, of 'moments of grace,' because the human being is able to experience in some way the Absolute who is utterly beyond" (n. 15). The role of "beauty as a key to the mystery and a call to transcendence" thus becomes a way to know God, though keeping in mind, as St. Augustine would remind us, that the beauty of created, finite things, cannot fully satisfy the human heart, but, instead, it triggers nostalgia for God (cf. n. 16). For the scientist, like for anyone else, beauty is what awakens enthusiasm for work, thus motivating his and her effort and research. It is precisely because of the relation beauty has with religious experience and with the access to God, that in this same letter John Paul II brings about again, and makes it his own, Dostoévskij's sentence, who, in his novel The Idiot, lets prince Myskin claim: "beauty will save the world."