Finding Beauty in Math and Science

When we try to attach a conceptual, encompassing definition of beauty, we start to trip over our feet. Beauty is inherently a disputed concept. What do we really mean when we say something is beautiful. Is it an objective quality, a subjective experience, a mathematical equation that can be measured and tested? Many arguments have been made throughout history regarding the nature of beauty. Not purporting what beauty ultimately is, I will draw upon a rich history of aesthetic theory to develop and support the argument of mathematics holding aesthetic qualities. It is not my position to advance a particular one here at this time, or argue between them all. It is my goal to demonstrate how when taking certain theories of beauty drawn form Plato, St. Thomas Aquinas, Kant, and others that an argument can be made about mathematics being “beautiful” and causing the “aesthetic experience.”

Introduction

Since the time of Plato, the meaning beauty has been debated. Various multidisciplinary theories on the meaning of beauty have been developed and deliberated. Whether beauty is based on subjective or objective characteristics still remains unsolved. The concepts of scientific and mathematical aesthetics are often considered as unrelated concepts. Initially, theories of aesthetics do seem to belong in the same realm of research as math. Alternatively, I will argue aesthetics in math is in deed a real area of research which has been propagated since the inception of aesthetical philosophy. Furthermore, both science and philosophy can advance research in aesthetics by finding common grounds on which to propagate their theories.

The primary focus of the aesthetics is upon the question of whether beauty is relative to the observer (beauty is in the eye of the beholder). The answer has a direct bearing on the practical problem of whether standards should be imposed upon the creation, appreciation, and criticism of art works. (Obitts, 2005). Aesthetic theories can be divided between our subjective awareness of beauty and the objective beauty of which we are aware. Aesthetic arguments may therefore focus either upon our ability to know beauty, or upon the existence of beauty itself. Aesthetic arguments that focus upon our knowledge of beauty are ‘epistemological’ arguments; those that focus upon the existence of beauty per se are ‘ontological’ arguments (Williams, 2001).

There is an inherent beauty in the functionality and simplicity of mathematical concepts. But, what is beautiful in math? In order to conceptualize beauty, it is necessary to outline the defining theories that shape the discourse of mathematical beauty. Primarily drawing form Plato, Kant, and St. Thomas Aquinas, we can see how the concept of beauty can be difficult to operationalize. Furthermore, in a discipline like mathematics, where the laws of concrete science dominate the ideological theories that are the basis of the discipline, we can find an inherent bias philosophical developments within the field of Aesthetics. Consequently, aggregating theories from various philosophical perspectives of aesthetics, this research will attempt to legitimize the theory of “mathematical beauty. “Examining current research on aesthetics, it is apparent that the enlightenment and the scientific revolution played vital roles in leading to the legitimization of mathematical beauty. Some example of how this will be achieved include:

âÂ?¢Through the ideal of truth, it will be demonstrated how mathematical concepts take on the characterization of beautiful; drawing upon Plato’s concept of truth is real beauty.

âÂ?¢Drawing upon the theories St. Thomas Aquinas it will further be argued how mathematics contains elements characterized by his theories that would classify it as “beautiful” through clarity.

âÂ?¢Augustine’s understanding of things to be “used” to bring us to blessedness can be applied to understand how math can bring us closer to true reality.

âÂ?¢Examining Formalism we can extract and apply the “Aesthetic Experience to mathematics.

Furthermore, drawing upon Huntley, the relationships between geometry and aesthetics will be explored. Specifically Huntley’s theory of how humans judge art as aesthetically pleasing, one is really looking for the breakdown of simple geometrical figures in arithmetical patterns (Huntley 136). In Addition, Huntley argues that the practice mathematics, provides an Aesthetic Experience described in Formalism. The concepts of scientific and mathematical aesthetics are often considered as unrelated concepts. Theories of aesthetics do belong in the same realm of research as math, and consequently science. Therefore, I will argue aesthetics in math is in deed a real area of research which has been proliferated since the inception of aesthetical philosophy.

Concepts of Beauty Related to Math

“Beauty is knowledge, but confused knowledge; or morality, but pleasant morality; or desire, but refined desire; or it is sentiment. It is truth- or again perfection- manifest to sense, the idea of the species, the individually characteristic; it is association, or unity in variety, or the je ne sais quoi; it is an inlikling of God’s beneficent creation, or a relaxation of our finer tissues” ( Carritt 1948:195).

Mathematics is outside and independent of us; it is discovered or observed not invented (Hardy, 1992). It is difficult to confine beauty to either objective or subjective categories. “It seems more satisfactory to regard it as an interaction between the mind and an object or an idea which arouses emotion”(Huntley, 1970:7). Recognizing beauty is indicative to some feature in the mind. Beauty is a word that has defied the efforts of philosopher to define in a way that commands general agreement (Huntley, 1970:11). Whether beauty is objective or subjective (or both) remains a metaphysical problem to be solved.

Traditionally in Western culture during Classical Times it was held that an affinity among three concepts (truth, goodness and beauty) existed. Plato formalized this notion (that Beauty was tied to Truth and to Goodness). Consequently, Plato advises the wise person to distinguish between apparent goodness, apparent truth, and apparent beauty because not everything that good is good, not everything that appears true is true, and not everything that appears beautiful is beautiful. Therefore, Plate though we should recognize their common beauty (form); recognize the beauty in the various kinds of knowledge and experience “beauty” itself (not embodied in anything, physical or spiritual.). This exemplifies a key characteristic of Platonic thought that knowledge rises through increasingly abstract levels.

Plato was the first to theorize on the subject of beauty. He believed that beauty was not an objective quality, but to be understood in the realm of ideas. Knowledge of reality is never changing and gained only through contemplation. This can be understood as facts are eternal and necessary. Plato argues that the beauty of shapes, for these are beautiful no with reference to anything else (Carritt, 1948:178). Plato sought the austere beauty of pure form and order. The absolutist view of the nature of mathematics has its roots in Platonism. Plato was concerned with “certain knowledge,” knowledge that is pure and abstract, not empirical or encumbered by social issues. (Brett, 2005). For Plato, mathematics was certain knowledge. Mathematical Platonism assumes that the existence of mathematics is independent of human beings or human activity. “It is the language of the universe” (Brett, 2005). Absolutism describes the nature of mathematics as infallible, unquestionable and unchanging objective truth based on a priori knowledge, revealed via reason only, and devoid of social context (Ernest, 1998). Therefore, for Platonists, mathematics would be classified as real beauty.

Furthermore, Plat outlines his views on beauty in the Symposium, and presents his view on the proper way to learn to love beauty:

1. Begin at an early age
2. First be taught to love one beautiful body (a human body).
3. Notice that the first body shares beauty with other beautiful bodies. (provides a basis for loving all beautiful bodies)
4. Realize that the beauty of souls is superior to the beauty of bodies.
5. (After the physical is transcended) then the second spiritual stage begins; learn to love beautiful practices and customs.
6. Recognize their common beauty (the form).
7. Recognize the beauty in the various kinds of knowledge.
8. Experience “Beauty” itself (not embodied in anything, physical or spiritual.)

Consequently, beauty was something to be understood as an ideal form, but not a concrete identifiable physical trait. Beauty for Plato is the clearest image of ideal truth. The visible beauty of the form and grace dispose men to right conduct according to Plato (Carritt, 1964:34). Therefore, taking this into account, math would be a great illustration of beauty. It was a means to knowledge or valuable truth. It was in the realm of abstraction, and additionally, it could be practically applied to the world.
Christian philosophy was heavily inspired by Platonic thought, although they did take a step away in their conceptualization of beauty in certain regards. Influential Christian thinkers like Saint Thomas Aquinas and Saint Augustine played vital roles in further legitimizing ideologies of beauty.

St. Thomas Aquinas’ philosophy expresses this empirical mindset, not an unworldly one, and combines both objective and subjective aspects. Forms are embodied in nature as we experience it and have no independent existence, a step away from the conceptualization brought forth by Plato. St. Thomas Aquinas’s Conception of Beauty Goes on to isolate the properties of the objects that do please and calm desire. Suggests three (objective) conditions, perfection, proportion or harmony, and brightness or clarity. Conditions of beauty are objective features. The idea of pleasing is a subjective element. Therefore, Aquinas combines the subjective and objective qualities of beauty. This represents a significant step away from the objective Platonic conception of beauty toward a subjective conception. As a result beauty is in essence correlated to being. What we perceive to be is what is given to experience and then conceptualized as a determined form. Consequently, mathematics seems to be an adequate conceptualization that can be operationalized from these concepts. It has the objective qualities or proportion and harmony, clarity, and perfection within the practice and understanding of the discipline, which ultimately leads to greater truth.

For St. Augustine there are not two worlds (as Plato held), but only one, and it is perfectly lucid. To enjoy something is to cling to it with love for its own sake. To use something, however, is to employ it in obtaining that which you love, provided that it is worthy of love. Therefore, real beauty can be associated with truth, as in this conception, true beauty would bring us closer to God. The things which are to be genuinely enjoyed are the Father, the Son, and the Holy Spirit (a certain supreme thing common to all who enjoy it). An illicit use should be called rather a waste or an abuse. Therefore, all things of value issue from the source of values -not the Form of the Good in this case, but rather “God. ” As a result, mathematics can be understood as something to be used in our course to blessedness from and Augustinian perspective; since utilizing mathematics we can come closer to true beauty.

Theories from the scientific revolution and the enlightenment took dramatic turns in their ways of conceptualizing beauty. Formalism came out of 18th & 19th century fascination with acquiring a rational/ scientific account of beauty (and other aesthetic qualities like sublimity). Formalism holds that there exists an unique class of human experiences which can be termed “Aesthetic Experiences.” It maintains that one should not judge art by its moral or educational work, but judge art by its own one of a kind aesthetic value. Therefore, art is to be appreciated for itself. So aesthetic experience is a value onto itself, and art is a vehicle for this aesthetic pleasure and that is all of its purpose.

The theory of Formalism is a way of approaching any and all works of art,. Furthermore, it is a vehicle for talking about and critically assessing any art in terms a formal and compositional understandings. This theory sets up art as a value onto itself. As the result of Formalism, we have a wide variety of new kinds of art forms; anything can be looked at in an aesthetic way. An aesthetic experience is an emotional (joyful and/or painful) and insightful experience from an interaction with anything. The next question is determining what relevance this definition has to mathematics (Betts 2005:10). Taking this into account, we can argue the practice of mathematics yields the “Aesthetic Experience.” Huntley (1970) elaborates on how completing mathematical formulas, one learns to appreciate math for math’s sake. Furthermore, the greatest appreciation of mathematics is attained from disinterested practice. When we are completing a math problem on a test, we are being hindered from the “Aesthetic Experience” because there is an interested motivation behind our work. When we enjoy mathematics for it’s own sake, we gain a greater appreciation and understanding of the formal properties involved with the elements of math, and reach closer to true beauty within the discipline of mathematics.

Kant further illustrates this disinterested observer notion critical to Formalism. Kant’s main objective was to show that the satisfaction derived from beauty is not as Hume, had maintained, sensuous, arbitrary, empirical and subjective. Kant attempts to distinguish pure beauty, from the beauty of things. “He is prepared for an aesthetic battle analogous to the one which he has fought for knowledge; and as that turned on the synthetic judgments a priori of pure mathematical intuition, so here he believes that the key situation is our judgment of pleasure universally communicable, yet independent of concepts” (Carritt, 1964:64). Kant’s general treatment of the aesthetic activity as a form of knowledge. The aesthetic activity is a becoming aware of our own inner nature and processes. Thus making us aware of things which were before obscure to us. It is epiphany moment, which gives us pleasures different from the pleasures of sense, because it is the first and always indispensable condition of spiritual activity (Carritt 1964:75). Schopenhauer, drawing from Kant, claims that everything in the world is capable of being found beautiful, perhaps in many different ways, if only we have the necessary genius (Carritt, 1964: 83). Kant also talks about being able to appreciate a purposeful-ness to things without being aware of their actual purpose. He has in mind here a certain “rightness” or “fitting-ness” we are aware of with great works of art. In other words, Kant wants us to appreciate an object disinterestedly. Therefore we do not care whether the object before us is real, imitation or wholly an illusion; as a result, we are indifferent to its actual existence.

“Social constructivism emphasizes the importance of culture and context in understanding what occurs in society and constructing knowledge based on this understanding” (Kim, 2001). Social constructivists outline two aspects of social context that largely affect the nature and extent of the learning, one of which is historical developments inherited by the learner as a member of a particular culture. Symbol systems, such as language, logic, and mathematical systems, are learned throughout the learner’s life. These symbol systems dictate how and what is learned. (Kim, 2001). The social constructivist view of the nature of mathematics is based largely on the philosophies of quasi-empiricism (e.g., Lakatos), conventionalism (e.g., Wittgenstein) and radical constructivism (e.g., Glasersfeld) (Ernest, 1991)(Brett, 2005). According to the research conducted by Lakatos (1976), there is a cycle of discovery in mathematics. This cycle begins with primitive conjecture, continues with proofs and refutations, and ends with a new beginning at an improved conjecture (Betts, 2005:12). This cycle suggests that mathematics is fallible since the creation of new mathematics is based on the discovery of flaws in previous conjectures. Further, deduction is not the logic of mathematical discovery (Lakatos, 1976).

Wittgenstein (1978) claims that certainty in mathematics is based on linguistic conventions, but these conventions change and evolve, so that mathematics is also continually in a state of change (Betts, 2005:12). Glasersfeld (1988) developed a theory of knowledge where the construction of knowledge is unique for each individual while still constrained by “viability” in the experiential world, and not the passive acquisition or discovery of an external and objective reality(Betts, 2005:13). Social constructivism is based on the theses that mathematics is fallible and changing, and that mathematical creation is by invention, not discovery of preexisting knowledge (Ernest, 1991). Mathematics does not exist independently of human beings (Betts, 2005:12). Social constructivism is in opposition to absolutism since mathematics is considered to be fallible, changing and the product of human inventiveness (Ernest, 1998).

Math and Beauty in Relation to Science

To assess the question of relevance of aesthetics in mathematics, the frameworks of production, criticism, historical/cultural place, and judgment will be utilized(Betts 2005:11). First, the act of doing mathematics (production) has an emotional component. From all the frustration, trial and error, dead ends and eventual success, there is a deep satisfaction from the process of searching for new knowledge (Stipek, 2002). Huntley (1970) also elaborates on the aesthetic experience within mathematics, and justifies the satisfaction derived for searching for new knowledge for knowledge’s sake. Second, every mathematician engages in a critique of mathematics when deciding that a theorem, conjecture or proof is beautiful. Hardy (1992) states that mathematics is beautiful and that there is no enduring ugly mathematics. Third, there are historical and cultural interactions with mathematics in terms of what is valued by mathematicians.

The production and justification of mathematics are embedded in personal, cultural and historical context (Lakatos, 1976). Finally, mathematicians make judgments concerning what is beautiful and of value in mathematics, and these judgments are based on assumptions concerning the nature of mathematics. In summary, mathematicians find mathematics to be beautiful and of value in a cultural and historical context by doing and understanding mathematics (Betts 2005:11). The value of mathematics is justified with the Platonic notion of real truth, and the practice of mathematics is justified by Aquinas’s understanding of clarity. Moreover, the Formalist understanding of “Aesthetic Experience” solidifies the search for knowledge for knowledge’s sake within the discipline.

Some of the earliest references to the pleasure of mathematics are linked to the Greek philosopher Pythagoras (Huntley, 1970:23). The explanation of the order and harmony of nature was, for Pythagoras, to be found in the science of numbers. The golden ratio, (also known as the divine proportion, golden mean, or golden section) is a number often encountered when taking the ratios of distances in simple geometric figures (Weisstein, 2005). The Greek mathematician Euclid (300BC) wrote the Elements which is a collection of 13 books on Geometry. In Book 6, Proposition 30, Euclid shows how to divide a line in mean and extreme ratio which we would call “finding the golden section G point on the line”.
< - - - - 1 - - - - ->
A G B
g 1-g
The result of the golden ratio equals 0.61803… and 1.61803âÂ?¦(Knott, 2005). “The term golden section seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik” (Livio 2002:6). The symbol (“phi”) was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002:5). The golden ratio is the most irrational number, yet it is apparent vastly through out our history and world. The proportions of the Greek Parthenon bear witness to the influence that the golden ratio had on their architecture. The Greeks were fascinated by the ratio, having it integrated throughout not only architecture, but artwork as well.

The claim to beauty in math at least in some areas of the discipline, is not built on an artificial basis but grounded in the beauty of the natural world (Huntley, 1970: 57). For example, the law of biological growth is exponential law. This is illustrated in the curvature of a shell, the mathematical formula to represent this equals a polar equation of an equiangular spiral.
It is generally accepted that there is a close association in mathematics between beauty and truth (Huntley, 1970:75).

Mathematical beauty serves a useful purpose as a guide to truth (Huntley, 1970:75). In addition, pleasure from the pursuit of “mathematics as a mental discipline springs from those deep layers of human psyche which, having developed in the early epochs of human evolution, lie beneath the mental strata of later development “(Huntely, 1970:78). Huntley argues that the ultimate source of aesthetic sensibility to the various manifestations of beauty in mathematics is to be sought for the unconscious mind or even in the collective unconscious by virtue of which man is the heir of all the ages (1970:79).

Furthermore, he identifies key ingredients to beauty in mathematics:

(1) alteration of tension and relief
(2) realization of a expectation
(3) the perception of unsuspected relationships
(4) sense of wonder and awe.

German psychologist, Gustav Feechner was the first to examine whether the Golden Ratio played part to special aesthetic interest. Feechner made thousands of ratio measurements of commonly seen figures, all of which were rectangles. After concluding his exploratory study, he found that the majority of people prefer the ratio of a rectangle closest to phi (which is related to the golden ratio). Fechner’s extensive experiments were made in 1876, and were rather crude, but have been repeated by various researchers and yielded similar results. Bilateral symmetry, and proportion seem to play a key role in the aesthetic experience. Feechner’s research exemplify the role that science can play in helping to understand the objective conditions that can elicit the “Aesthetic Experience.” Illustrating how Aesthetics is a cross disciplinary field, not limited to the realm of philosophy. Moreover, how mathematics is a crucial element in the “Aesthetic Experience.”

Another interesting aspect of aesthetic quality of math can be found in the realm of statistics. There is a tendency to try to acquire data as close as possible to the normal curve because of the symmetry of the curve. It is a mirror proportion to itself, complete symmetry. It is not only statistics which finds symmetry ideal in shapes, but geometry, trigonometry, and algebra also. There is a magnitude of research which suggests that it may be biological hardwiring within our brains that leads us to find symmetry aesthetically pleasing; thus accounting for the tendency to lean towards such forms and figures as beautiful. Symmetry was found to be the most important stimulus feature determining

participants’ aesthetic judgments in various scientific research endeavors (Jacobsen et al., 2004:5). This further supports the argument for biological hardwiring, or neurological preferences for certain symmetrical shapes and objective qualities in art, that are dictated by mathematical laws. Further developing the interwoven relationship that mathematics shares with the “Aesthetic Experience.”

It has been well-recognized that aesthetic criteria play a powerful role in determining the design of theoretical models as well as the dynamic equations of physics. The problem of insight in theory-building, problem-solving, and reasoning generally has been tackled with significant advances in the problem-reformulation community, which is based strongly on the aesthetic supervision of discrete algebraic systems (Leyton 2005). Scientific advance have often relied on aesthetic qualities of theory development, especially with in the realm of advanced mathematics.

Conclusion

“Born of man’s primitive urge to seek order in his world, mathematics is an ever-evolving language for the study of structure and pattern. Grounded in and renewed by physical reality, mathematics rises through sheer intellectual curiosity to levels of abstraction and generality where unexpected, beautiful, and often extremely useful connections and patterns emerge. Mathematics is the natural home of both abstract thought and the laws of nature. It is at once pure logic and creative art” (White, 1993:5).

Even with St. Thomas’ (small) move away from the Metaphysical Explanation of Beauty, Western thought was slow to give it up entirely.

Even into contemporary times we see those who suppose “beauty” is a window to a different “higher” realm of reality. Discursive analysis of research history on the Aesthetics, we see to critical factors: 1) the power level of the theory (who pushes, supports they theory publicly in society; 2) ideological shifts in society can either bring about greater research emphasizes on theories that are dominant while pushing competing theories to the side. Therefore, a competing theory must have public support by powerful individuals or institutions to bring about an ideological shift in research. In order to reach a point where both science and philosophical Aesthetics research to advance to a greater understanding, common ground must be exploited through theoretical frameworks. Mathematics serves as the basis for such an exploitation. Mathematics has been proven both in Scientific Aesthetic research, and Philosophical Aesthetic research to play an instrumental role in the Aesthetic Experience.

Utilizing Wittgenstein metaphor of family resemblance, we can aggregate objective qualities of beauty to perform adequate research on the nature of beauty to advance the discipline of Aesthetics in our modern era. If we gather together five members of the same family, they probably look alike, although there is no distinctive feature that they all share in common. A brother and a sister might have the same dark eyes, while that sister and her father share a slightly turned-up nose. They have a group of shared features, some of which are more distinctly present in some members of the family, while some features are not present at all (Sparknotes, 2005). Wittgenstein argues that the different uses of one word share the same family resemblance. There is no single defining characteristic of all uses of the word “understanding”; rather, these uses share a kind of family resemblance with one another. That is the inherent obstacle that research into beauty has faced. What is the true meaning of beauty. As Aesthetical research pushes forward through various disciplines, we have been able to identify a group of shared features that characterize the Aesthetic Experience. Of which, a dominant reoccurring feature has been mathematics, crucial in every aspect of theoretical developments of beauty. Consequently, the development of objective conditions that lead to the Aesthetic Experience have to delve into the understanding of the mathematical relationships present in the various forms of art, and various other mediums, that elicit the Aesthetic Experience.

Appendix

A beautiful equation: illustrating the “Aesthetic Experience,” math for math’s own sake.

eip + 1 = 0

This equation is true (trust me!), but how is it beautiful? Let’s examine the numbers
involved.

0: Add 0 to anything and you get that number back. It is the number that somehow means
“nothing.”

1: Multiply any number by 1 and you get the number back. It the number you start at when you count.

p: This is the ratio of the circumference of a circle to its diameter, about 3.14. It originates in these geometric areas of mathematics.

e: This is a number you have seen if you have had calculus (the derivative of ex is ex, which makes it special), and it is about 2.718. It is very important in mathematics like calculus.

i: This is the square root of negative one. Even though it is often called “imaginary,” it is very important in real areas of mathematics. These might be considered the five most important numbers in mathematics. 0 and 1 are from basic arithmetic, p gets its start in geometry, e is used in calculus, and i is not even a “real number,” but all five are combined into this one simple equation. This equation unites areas of mathematics that, at first glance, do not seem related at all! The startling simplicity with whichthis equation ties mathematics together is the reason for its beauty.

(Source: Woods, 2005)

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