Donât become a mere recorder of facts, but try to penetrate the mystery of their origin.
ivan pavlov (1849â1936)
Talk of mysteries! Think of our life in nature, daily to be shown matter, to come in contact with it, rocks, trees, wind on our cheeks! The solid earth! The actual world! The common sense! Contact! Contact! Who are we? Where are we?
The word mystery comes from the ancient Greek mustÄrion, designating a âsacred secret.â It was applied to characterize the ancient polytheistic cults (Hall 1973, Mishlove 1993, Bremmer 2014), of which the one founded by the Greek mathematician and philosopher Pythagoras (c. 580â500 bce) was the most famous.
The central objective of the Pythagorean cult was to investigate mathematics as a code that would help them unlock the secrets of the universe. Pythagoras had discovered that plucking strings or striking objects in a certain way produced harmonious sounds that could be described with specific numerical ratios. From this, he reasoned that the planets moving in orbits also produced them, being part of the same physical worldâa view that musical harmonies informed the structure of the universe, which came to be known as the theory of musica universalis (literally, universal music), or the âmusic of the spheres.â The harmonious sounds emitted by the planetary orbital revolutions are imperceptible to the human ear, unlike human musical sounds which are audible. Legend has it that Pythagoras could actually hear the celestial soundsâa gift conferred on him by the Egyptian god Thoth. This is, of course, part of the lore that surrounded Pythagoras. What is likely true is that Pythagoras derived his musica universalis theory from the Egyptians and the Chaldeans who held that the celestial bodies emitted a âcosmic chantâ as they moved through the sky (Burkert 1972). Even in the Bible (Job 38.7), the beginning of time is described as the moment âwhen the stars of the morning sang together and all the sons of God raised a joyous sound.â
Contrary to traditional belief that a biological system is either at stead-state or cycles with a single frequency, it is now appreciated that most biological systems have no homeostatic âset point,â but rather oscillate as composite rhythms consisting of superimposed oscillations. These oscillations often cycle at different harmonics of the circadian rhythm, and among these, the ~12-hour oscillation is most prevalent ⦠We posit that biological rhythms are also musica universalis: whereas the circadian rhythm is synchronized to the 24-hour light/dark cycle coinciding with the Earthâs rotation, the mammalian 12-hour clock may have evolved from the circatidal clock, which is entrained by the 12-hour tidal cues orchestrated by the moon.
The key mathematical notion in musica universalis is that of commensurable ratios. What the Pythagoreans did not anticipate, however, was their unwitting discovery of numbers that were incommensurable, subsequently called irrational. So disturbed were they by this that, according to legend, they âwent to the length of killing one of their own colleagues for having committed the sin of letting the nasty information reach an outsiderâ (Ogilvie 1956: 15). The colleague is suspected to have been Hipassus of Metapontum (Bunt, Jones, and Bedient 1976: 86). While this is likely to be a myth, the fact remains that the discovery of irrational numbers was mind-boggling, going (purportedly) against the musica universalis view of the cosmos.
Pi (Ï), or the ratio of the circumference of a circle to the diameter, was one of the first irrational numbers discovered. As such, it constituted a conundrum with regard to musica universalis. Pi represents a stable pattern in physical structureâit stands for the inescapable fact that, as the circumference of any circle increases, so too does its diameter, in a proportional way. As such, Ï would seem to fit in perfectly with the Pythagorean view of harmony, revealing an underlying pattern to an aspect of physical reality based on proportion. But the fact that Ï turns out to be an irrational number poses an existential paradox to the Pythagorean model of the world: Why is an incommensurable number connected to one of the most harmonious of all geometric forms, the circle? The mystery deepens by virtue of the fact that Ï has cropped up in various mathematical formulas that describe biological and physical structures and processes which, on the surface, seem to have nothing to do with circles.
So much has been written about Ï that it would be presumptuous to suggest that the themes I will be discussing in this book are novel (for comprehensive treatments see Beckmann 1971, Blatner 1997, Eymard and Lafon 2004, Posamentier 2004). Rather, my goal is to revisit the geometry of the Pythagoreans and its foundation as a âhermeneutic science,â that is, a method of inquiry aiming to investigate the connectivity between geometric figures, numbers, and reality. The term hermeneutics (from Greek á¼ÏμηνεÏÏ, hermÄneuÅ) was introduced into philosophy by Aristotle in his book Peri Hermeneias, translated into Latin as De Interpretatione, and later in English as On Interpretation (Aristotle 2016). There is some suggestion that it may have a pre-Greek origin (Beekes 2009: 462). Whatever the case, it was Aristotle who made the cogent argument for the need to develop a philosophical method of âinterpretationââthat is, a method for explaining the meanings of things. The geometry of the Pythagoreans can be retroactively described as hermeneutic, since they believed it would provide a key to decoding the meaning of the universe.
In the medieval period, the term hermeneutics designated specifically the interpretation of sacred scripture (Grondin 1994: 21); it continues to be used with this designation within theology to this day. Starting in the nineteenth century, the same term surfaced in philosophy as a theory of understanding, as can be seen in the writings of Friedrich Schleiermacher, Wilhelm Dilthey, Martin Heidegger, Hans-Georg Gadamer, among others (Seebohm 2007, Zimmerman 2015). It was not until the twenty-first century, however, that the term migrated to the domain of science and mathematics, where it is now used to identify any approach that connects the study of mathematics with humanistic disciplines, such as music and the arts. Although they did not identify it as such, George Lakoff and Rafael Núñez put forth one of the first comprehensive hermeneutic treatments of mathematics in their 2000 book, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (2000). Lakoff and Núñez explained the ontological basis of mathematics as reflecting the same cognitive structures that underlie language, and especially metaphorical language. Since then, the hermeneutic approach in mathematics has spread broadly, especially in the educational sphere where it has had significant applications (see, for instance, Danesi 2019).
De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive; and quoted the actuarial formula, involving Ï [pi], which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, âMy dear friend, that must be a delusion, what can a circle have to do with the number of people alive at a given time?â
Marcel Danesi
University of Toronto, 2020