This week, I would like to examine the idea of beauty. First, I would like to think about the definition of beauty. In Bel Canto, it seems that the time which is spent listening to Roxane Coss sing is the only time that the characters truly live. This is one definition, certainly, the idea that as people feel beauty they also feel a connection to themselves and their lives cannot be discredited. However, I would like to think of a more empirical idea of beauty. In his book, H. E. Huntley refers to beauty with the idea that the experience is the source of the pleasure (12). Huntley argues that the discovery, the unearthing of a mystery, and the explosive result is the true beauty in the world, and that that beauty can be found through mathematics (12). In an article by Matthew Inglis, it is argued that “mathematicians perceive simplicity to be a characteristic of beautiful mathematics.” (91) Inglis goes on to posit that, “’beauty,’ unlike ‘enlightenment,’ is a concept which does not admit degrees.” (92) The thought that mathematicians have an idea of beauty as it is applied to music interests me, as I can imagine that there are many ways in which it could be applied towards the rest of the world, creating an intriguing overlap of empirical and emotional views.
In a previous week, I defined the Golden Ratio (phi), as 1.61803… onwards. This fact should be kept in mind, as it will continue to come back up in the discussion of math as beauty. Huntley explains that “a rectangle, the lengths of the adjacent sides of which are in a ratio which is exactly or close to phi:1, appears to afford a greater measure of satisfaction” than others (52). According to Huntley, this can be related to music and rhythm. In the discussion of Fibonacci numbers and the golden ratio previously, it was noted that the ratio of frequencies of certain intervals reduced to the golden ratio. We find that the ratio of frequencies in a major sixth in music is 8:5 (Huntley 54). With a small amount of math, we see that 8/5=1.6, and phi=1.61803…, making the two ratios comparable. This begins to explain some of the attraction towards certain musical intervals, but fails to shed any light on the reasons as to why we find this relationships appealing. This is one instance in which we see that mathematical understanding is not necessarily key to grasping the relative beauty of the formal depiction of an idea. In an article by George Markowsky, it is argued that the idea that phi is evinced in many different places in the manmade world is incorrect (1). He does not argue that the number is not in evidence in the natural world, merely that our perceptions about the historical and artistic uses of the relationships therein are misconceptions. I think that this counterargument, the discussion of what is related to phi as opposed to what is not related to phi, could become an entire research discipline in and of itself.
But what is beauty? How can we define beauty in such a way that it makes sense to the entirety of the world? From experience, we could each say that beauty is a personal ideal, some will have different ideas as to what is beautiful than others, but is there some overarching idea that ties it all together? That is a question that, I believe, needs to be answered in order to construct a meaningful conversation about the definition of beauty. Rhett Diessner (et al.) address three different kinds of beauty, natural, artistic, and moral (302). Moral beauty is defined as “the expression of any of these virtues” (Diessner 304). The virtues in question are “Wisdom and knowledge, courage, humanity (love and kindness), justice, temperance, and transcendence.” (Diessner 304) We also see that Huntley refers back to the “importance of the emotions that beauty calls forth.” (11) He expands on this by explaining that beauty should bring about great emotion, but he also makes a note that not everyone will react the same way to beauty (11). Therefore, I think that one of the greatest challenges in defining beauty and exploring the idea of beauty in mathematics is in simplicity – it seems to me that there is an innate appreciation for reduction and clarity, and that perhaps that itself is a universal kind of beauty that can be evoked by the use of standard, innately appealing ratios.
Inglis, M, and A Aberdein. “Beauty Is Not Simplicity: an Analysis of Mathematicians’ Proof Appraisals.” Philosophia Mathematica. 23.1 (2015): 87-109. ArticleFirst. Web. 7 Oct 2015
Huntley, H E. The Divine Proportion: a Study in Mathematical Beauty. New York: Dover Publications, 1970. Print.
Diessner, Rhett, Teri Rust, and Rebecca C (& others) Solom. “Beauty And Hope: A Moral Beauty Intervention.” Journal Of Moral Education 35.3 (2006): 301-317. Philosopher’s Index. Web. 7 Oct. 2015.
Markowsky, George. “Misconceptions about the Golden Ratio.” The College Mathematics Journal 23.1 (1992): 2-19. Google Scholar. Web. 7 Oct 2015.