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Theoretical : 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 , 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 mathematics, it is primarily to introduce great

Greek of the Classical Period and their achievements1: Zeno of Elia (490-430

B.C.), of (470-410 B.C.), of Elis (born about 460 B.C.), (427-

347 B.C.), (390-337 B.C., according to Wikipedia2), and (384-322

B.C.).

When his next ten lecture notes (11-20) (2016) discuss Ancient , it is primarily to introduce great Greek mathematicians of the ; of

Syracuse (287-212 B.C., lecture 12), of (about 284-200 B.C., lecture 13),

Apollonius of (262-190 B.C., lecture 11); review the broad contributions and limitations of

1. Only (624-547 B.C.), of (born between 580-572, died 497 B.C.), and (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 (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 , 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 () concrete objects] pervasive in society. Asper writes that it is false to conceive of “theoretical mathematics” (the type concerned with -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 ”. 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 were primarily interested in measuring 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 of having a leisure class.

Though we think of ancient 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 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 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.

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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 with its 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 “” 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 ”, 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”.

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Verb 2: The three famous geometric construction problems of Ancient Greece originated with the men of practical math, but the men of leisure took an interest in them. This was the origin of theoretical math: men of leisure taking an interest in the three famous problems.

Asper theorizes that the three great geometric problems of Ancient Greece started in the practical mathematics realm of the craftsmen. He believes this because they are all procedure-based problems, asking for a procedure as the solution. For us, with the weight of history behind them, we might think of these problems as imposing. But for the men of leisure, Asper characterizes these problems as “riddles” (2009, p.122). In modern terms, we might refer to them as

“brainteasers”.

We can imagine that when members of the leisure class were exposed to these riddles, they had such time and energy to devote to them as no craftsman ever had. Let us put the half-mythical beginnings of theoretical mathematics aside. The first theoretical mathematics text— i.e., in the axiom-based, deductive lineage that traces through Euclid’s Elements to modern times—is by

Hippocrates of Chios (Asper, 2009). And its subject is the of lunes, related to squaring the .

But why did the members of the leisure class attack the three problems with such gusto that they developed axiom-based deductive proof? My own thinking is that a precursor form of Plato’s philosophy (perhaps espoused by his teacher , or even earlier) played a role in predisposing the members of the leisure class toward appreciating abstract objects. So when they were introduced to the three problems, they found something aligned with their interests.

Play

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Now you know Asper’s theory of the origin of theoretical mathematics. If you were to characterize the Verb 1 and Verb 2 sections with one word, what would you choose? Asper chooses “play”, and I wholeheartedly agree.

According to the great cultural historian Johan Huizinga, the chief distinction between work and play is that work has real-life consequences, but play doesn’t and so it is carefree (Huizinga,

1980). (He acknowledges you can make play into work, but then it is no longer true play.)

For the Ancient Greek leisure class, spending their time on theoretical mathematics had no impact on their fortunes, and only later impact on their status. For them—at least in the early days—it was play.

On Status and Flourishing

Now we move from inception to spread. Why did the leisure class study the geometric objects in the three problems un-tethered to any procedures (the original riddle requirement)—simply listing and proving properties of those objects? Why did they move on to studying other geometric objects unrelated to the three problems? In other words, beyond how theoretical mathematics was born—why did it flourish from birth?

From an ethically positive standpoint, my thoughts on Plato’s philosophy, which I have already introduced, offer an explanation.

However, Asper offers another explanation, which happens to be ethically negative, and which I find very compelling: that theoretical mathematics flourished (at least partly) because it provided the tangible benefit of being a social status signifier.

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Can we agree that it makes sense that any activity that only a member of the wealthy elite can engage in, might become a status signal that you are a member of that wealthy elite? For example, yachting is something only a wealthy person could do, so yachting could be a status signal that a person is wealthy (and it is). Well, in Ancient Greece, theoretical mathematics was something only the wealthy elite could engage in. In Athens, the barrier to school attendance was not the cost of tuition. Rather, it was the opportunity cost of school attendance. Only a young man from a wealthy family could forgo the income of becoming an apprentice in a trade

(Guisepi, n.d.).

Further, theoreticians actively distanced themselves and their activity from any practical, monetary benefit. Asper offers an anecdote about Euclid that is likely fictional, but communicates the prevalent attitude of the theoreticians: “Someone who had taken up geometry with Euclid, asked after he had understood the first theorem: ‘What is my profit now that I have learned that?’ And Euclid called for his servant and said: ‘Give him a triōbolon, since he must always make a profit out of what he learns’” (Asper, 2009, p. 124).

With no income as a student of theoretical mathematics, and no income as a practitioner of theoretical mathematics, then only the wealthy could practice theoretical mathematics.

I would like to conclude with the words of two of the greatest mathematicians of Ancient

Greece, greats among greats: Apollonius and Archimedes. When I learned they sometimes wrote introductory letters when sending their proofs to friends, I was curious what sense of the men I could make from their regular, first-person writing.

“Apollonius to Eudemus, greeting: ‘If you are in good health, it is well. I too am modestly well. I have sent my son Apollonius to you with the second book of my collected conics. Peruse it

8 carefully and communicate it to those who are worthy to take part in such studies (Apollonius,

1896, Preface to Book II, p. lxxii).”

“Now, however, it will be open to those who possess the requisite ability to examine these discoveries of mine. They ought to have been published when Conon was still alive, for I should conceive that he would best have been able to grasp them and to pronounce upon them the appropriate verdict; but, as I judge it well to communicate them to those who are conversant with mathematics, I send them to you with the proofs written out, which it will be open to mathematicians to examine (Archimedes, 1897, On the and the Cylinder, Book I, p. 2).”

Don’t their words drip with exclusivity and a desire to maintain that exclusivity? Sigh, they seem not the heroes I had hoped for.

References

Alexander. (n.d.). In SparkNotes. Retrieved October 9, 2016 from http://www.sparknotes.com/biography/alexander/section2.rhtml

Ancient Greece. (2016). In Compton's by Britannica. Retrieved October 9, 2016 from http://kids.britannica.com/comptons/article-201729/ancient-Greece

A., & Heath, T. L. (1896). : on conic sections. Cambridge:

Cambridge University Press.

A., & Heath, T. L. (1897). The works of Archimedes. Cambridge: Cambridge University Press.

Aristotle. (n.d.) In Wikipedia. Retrieved October 9, 2016, from https://en.wikipedia.org/wiki/Aristotle

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Asper, M. (2009). The two culture of mathematics in Ancient Greece. In The Oxford Handbook of the History of Mathematics (pp. 107-132). Oxford: .

Davy Crockett. (n.d.). In Wikipedia. Retrieved October 9, 2016, from https://en.wikipedia.org/wiki/Davy_Crockett

Guisepi, R. (n.d.). The history of education. Retrieved from http://history- world.org/history_of_education.htm

Huizinga, J. (1980). Homo ludens: A study of the play-element in culture. London: Routledge &

Kegan Paul.

Ji, S. (2016). Lecture 18. Mathematics in Medieval Europe [PDF document].

Martínez, A. A. (2012). The cult of Pythagoras: math and myths. Pittsburgh, PA: University of

Pittsburgh Press.

Plato. (n.d.) In Wikipedia. Retrieved October 9, 2016, from https://en.wikipedia.org/wiki/Plato

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