Theoretical Mathematics: Born of Play by the Leisure Class, Flourished as a Social Status Signal By Allen Chai In his lecture notes, Dr. Ji states that the history of Ancient Greece, as a civilization, can be divided into four periods, and he gives specific start and end dates: the Greek Dark Ages (1100- 750 B.C.), the Archaic Period (750-480 B.C.), the Classical Period (500-323 B.C.), and the Hellenistic Period (323-146 B.C.). When his first ten lectures discuss Ancient Greek mathematics, it is primarily to introduce great Greek mathematicians of the Classical Period and their achievements1: Zeno of Elia (490-430 B.C.), Hippocrates of Chios (470-410 B.C.), Hippias of Elis (born about 460 B.C.), Plato (427- 347 B.C.), Eudoxus of Cnidus (390-337 B.C., according to Wikipedia2), and Aristotle (384-322 B.C.). When his next ten lecture notes (11-20) (2016) discuss Ancient Greek mathematics, it is primarily to introduce great Greek mathematicians of the Hellenistic Period; Archimedes of Syracuse (287-212 B.C., lecture 12), Diophantus of Alexandria (about 284-200 B.C., lecture 13), Apollonius of Perga (262-190 B.C., lecture 11); review the broad contributions and limitations of 1. Only Thales of Miletus (624-547 B.C.), Pythagoras of Samos (born between 580-572, died 497 B.C.), and Euclid of Alexandria (lived around 300 B.C.) don’t belong to the Classical Period. Thales and Pythagoras belong to the Archaic Period, but it is reasonable to question their inclusion as great mathematicians, or mathematicians at all. There have been serious doubts raised about whether Pythagoras, let alone Thales who preceded him, actually practiced mathematics (Martinez, 2012). A handy descriptive term for the pair would be “half-mythical” (Asper, 2009). What has happened to the two is a psychological phenomenon that has occurred throughout history, such as with Davy Crockett “King of the Wild Frontier”. It is the phenomenon of real men who in life are already credited with larger-than-life acts; then after they die, new and even more outlandish stories continue to be created (“Davy Crocket”, n.d.). Euclid belongs to the Hellenistic Period, which Dr. Ji focuses on in his second series of lectures. 2. Unless otherwise specified, dates are from Dr. Ji’s notes. I relied on Wikipedia for Eudoxus, because Dr. Ji’s notes give his lifetime as 480-355 B.C., which doesn’t make sense from a historical or longevity standpoint. Greek mathematicians across the historical periods of Ancient Greece (lecture 14); and to introduce reasons why mathematics was born and flourished in Ancient Greece and not elsewhere (lecture 18). It is this last topic that I intend to develop today. Please allow me to nerd out for a moment: In a famous episode of Star Trek: The Next Generation called “All Good Things…” the omnipotent entity known as “Q” takes Captain Jean- Luc Picard back in time to the beginning of life on Earth. The inception of life on Earth is depicted as the oozing of a collection of primordial goo (ingredient 1) into another collection of primordial goo (ingredient 2), voila resulting in life. Now compare this with Dr. Ji’s passage that most relates to why theoretical mathematics was born and flourished in Ancient Greece: “Mathematics has been most successful in a free intellectual atmosphere [ingredient 1] in which there are some people interested in the problems presented by the physical world [ingredient 2] and there are some people interested in thinking about abstract problems arising from practical problems that make no promise of immediate return [ingredient 3]” (Ji, 2016, p. 115). Do you feel the same knowledge itch that I do? In both instances, we are told of the nouns (“primordial goo”, or “a free intellectual atmosphere” and “some people interested…”, etc.) that are the ingredients, the parts, that form a final product (“first life on Earth”, or “theoretical mathematics”) that is a sum undeniably greater than the parts. But the entire story—the verbs, the actions, that involve the nouns—of how the parts became that greater-than sum is missing. We’re going from raw ingredients to meatloaf, completely skipping any cooking directions. We don’t know how the onions became meatloaf. And we don’t 2 know how a “free intellectual atmosphere” and “some people interested” in practical problems and “some people interested” in abstract problems became theoretical mathematics. And that is what I hope to contribute today. A coherent theory—the nouns and verbs—of how theoretical mathematics developed in the world. The originator of this theory is Dr. Markus Asper, a professor of Greek at the Humboldt University of Berlin. The challenge with Asper’s writing is that it isn’t always clear. In parts, I am either presenting his ideas more clearly than he does (to the benefit of the reader), or I have gotten off the road and am presenting new and unproven ideas. Therefore, for those truly interested in the development of theoretical mathematics, please read his text “The Two Cultures of Mathematics in Ancient Greece” found in The Oxford Handbook of the History of Mathematics, and use my essay as a companion. Ingredient 1: Practical mathematics pervasive in society In Ancient Greece, you already have “practical mathematics” [the type concerned with counting (multitude) and measuring (magnitude) concrete objects] pervasive in society. Asper writes that it is false to conceive of “theoretical mathematics” (the type concerned with axiom-based, deductive proofs involving abstract objects) as the only mathematics practiced in Ancient Greece: “Recently, however, a consensus has emerged that Greek mathematics was heterogeneous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices” (2009, p. 107). Asper writes that the repository of practical mathematics knowledge was the professional class: craftsmen, tradesmen. For example, in counting, there was a group of professionals who engaged in “pebble arithmetic”. They used pebbles as “psēphoi”, (2009, p. 108) or “counters”, on a 3 marked surface (think of it as a Western abacus) to perform various arithmetic calculations. In measuring, the Ancient Greeks were primarily interested in measuring areas and volumes. One group of professionals were the “harpēdonaptai”, (p. 113) which literally means “rope- stretchers”, who were surveyors that used ropes to do measuring. Ingredient 2: Presence of a leisure class In Ancient Greece, you also have a society “developed” to the point of having a leisure class. Though we think of ancient Athens as the birthplace of democracy, we’d do well to remember it was highly stratified with estimates of a slave population at between 40-80 percent of the total population (“Ancient Greece”, 2016). Asper writes of the “upper circles of Athenian society” (2009, p. 123). I prefer the term “leisure class”. For I believe the key characteristic of these elites as it pertains to the development of mathematics was not their wealth and power, but what that wealth and power afforded—time free of any obligations. When records of their early lives exist, the great Ancient Greek mathematicians are revealed as members of the upper circles, of the leisure class. For example, Plato “came from one of the wealthiest and most politically active families in Athens” (“Plato”, n.d.). Aristotle’s father “was the personal physician to King Amyntas of Macedon” (“Aristotle”, n.d.). Ingredient 3: A unique philosophy that favors abstract objects, and disfavors concrete objects Asper doesn’t write about Plato’s philosophy of the Theory of Forms as a factor in the development of theoretical mathematics. However, I see its imprints in the activity and attitudes of Ancient Greek mathematicians. 4 What exactly Plato’s philosophy was, is a question that leads to a spider web of scholarship, to which you can lose your life studying. Very broadly, he believed that there is a reality of eternal, abstract objects separate from the physical reality of concrete objects available to our senses (“Plato”, n.d.), and that the former reality is superior to the latter. If a philosophy favors abstract objects, and disfavors concrete objects, then—logically—its adherents would display the same pattern of favor and disfavor. And throughout the history of Ancient Greece, we see the great mathematicians displaying this pattern. By their very activity on mathematical objects, they show their favor for abstract objects. Further, some of them, such as Aristotle, are famous for their contempt of hedonism with its focus on concrete objects, and preached self-denial and self-control (“Alexander”, n.d.). Verb 1: The men of leisure interacted with the men of practical math, and/or interacted with their practical math knowledge to a degree. The evidence for this interaction is in the language. If theoretical mathematics originated in isolation from practical mathematics in Ancient Greece, then why would the two realms share so much of the same terminology? Why would the craftsman’s Greek word for “angle” be the same as the theoretician’s Greek word for “angle” (Asper, 2009, p. 122)? If there were truly a state of isolation, the theoretician would have created or selected a novel term. The evidence moves past the names of mathematical objects. Why would theoreticians use words uncommon to regular life, but common to craftsmen? For example, when Euclid describes “drawing a straight line”, he uses the word “teinō”, which literally means “stretch out”. Asper (2009, p. 122) posits he would only choose such a term if he, or an earlier theoretician that established the precedent, was influenced by the “harpēdonaptai”, the “rope stretchers”.