Switching Camps in Teaching Pythagoras By Allen Chai I typed Pythagoras into a search terminal in the M.D. Anderson Library. Is Pavlovian the right word to describe the way that name springs to top-of-mind at the mere mention of Greece and mathematics? In any case, by far the most interesting-sounding search result was Alberto A. Martinez s The Cult of Pythagoras: Math and Myths. I was pleased when I saw how new and clean the book looked compared to its mustier brethren on the shelf. But soon after I cracked the cover, that pleasure went away: Almost immediately I found contradictions with our Lecture 4. Pythagoras Theorem and the Pythagoreans notes. Which was I to believe? Our lecture notes, or the book? I was distressed. At this point let me say I didn t just crack the cover of Martinez s book. When you physically hunt for a call number, you must read many call numbers to close in on your target. And you can t help but to glance over the spines. On the third floor of the Blue Wing, I stumbled upon an entire bookshelf of books devoted to the history of math. I started to browse if something stood out about a book, such as a recent publishing date, or an intriguing or authoritative title, I would pluck it from its shelf. What I found is that writers generally present Pythagoras to readers in one of two ways: May Have Camp: A large contingent of the May Have Camp writers have noticeable grammar-logic inconsistencies in their writings about Pythagoras. Let me explain:
Many of this camp s writers present a big picture view of Pythagoras as a mathematician and his mathematical contributions as a whole, with the grammar of certainty. The language takes the form of: Pythagoras WAS a great mathematician. He MADE important mathematical contributions. According to Betty Azar, the author of numerous books on English grammar, the verb be on its own represents 100 percent probability, i.e., certainty (Azar, 2006). E.g., Pythagoras was a man. But, and this is a big but, when these writers present a zoomed-in view of any particular mathematical contribution attributed to Pythagoras, say his status as the first to prove the Pythagorean theorem, they change to the grammar of uncertainty. The language takes the form of: Pythagoras MAY HAVE MADE important mathematical contribution A, B, C Returning to Betty Azar, she says modal verbs such as may have and might have represent an about 50 percent probability which reflects uncertainty (Azar, 2006). E.g., Pythagoras may have been a man. The grammatical inconsistency clearly on display between these writers big picture views and their zoomed-in views is a symptom of an underlying logic inconsistency. How do uncertain antecedents, may have made important mathematical contribution A, may have made important mathematical contribution B, may have made important mathematical contribution C produce a certain consequent made important mathematical contributions? This doesn t make sense, hence the logic inconsistency. And if we cannot say that Pythagoras made important mathematical contributions, then it follows we cannot say he was a great mathematician.
We can see Ji s notes contain this grammar-logic, certainty-uncertainty inconsistency. In the introductory paragraph, Ji presents a big picture view of Pythagoras and writes he made important contributions in mathematics, astronomy, and the theory of music (Ji, 2016, p. 23). Yet when Ji presents a zoomed-in view of the particular mathematical contribution of first proving the Pythagorean theorem, he writes: Pythagoras s theorem was known to the Babylonians 1000 years earlier but Pythagoras may have been the first to prove it (Ji, 2016, p. 23). [I am restricting my analysis to mathematics, but there is little evidence Pythagoras contributed anything to the sciences either (Martinez, 2012).] Ji is far from alone in this grammar-logic, certainty-uncertainty inconsistency. Eli Maor does the same. For example, with his The Pythagorean Theorem: A 4,000-Year History, Maor, according to Martinez (2012, p. 45), glorifies Pythagoras, but demurs from this glorification once he zooms in to present particular achievements. In the synopsis for the book, Maor writes in regard to Pythagoras proving the Pythagorean theorem: Although attributed to Pythagoras, the theorem was known to the Babylonians more than a thousand years before him. He may have been the first to prove it, but his proof if indeed he had one is lost to us (Maor, 2007). We can guess why such inconsistencies exist. Likely, these writers were ruled by competing desires. There are a variety of reasons for presenting Pythagoras as a great mathematician, including the desire to maintain a mathematics hero figure (Martinez, 2012). Yet there is also the desire to cover your own butt when faced with a dearth of evidence of Pythagoras s mathematical greatness. In any case, there is a quick fix to these inconsistencies. Simply take the modal auxiliary verbs already employed in the zoomed-in views and also employ them in the big picture views.
This yields: Pythagoras MAY HAVE BEEN a great mathematician. He MAY HAVE MADE important mathematical contributions. The end result would be writing similar to that of former student Liu s Hippasus and Evolution sample essay. Observe Liu s (p. 1-2) passage on Hippasus. He entreats us to: imagine what it was like for Hippasus who is credited with proving that the square root of two could not be a rational number Hippasus colleagues in the Pythagorean Brotherhood had searched in vain for the two integers that made up the numerator and denominator of the fraction. Imagine being told that it was crazy to have spent all that time! it comes as no surprise that Hippasus may have been murdered: and so they drowned him in the time-honored tradition of killing the messenger. Maybe. One source claims this story is unlikely Liu maintains strict consistency in his use of modals and caveats throughout the passage. Any piece of information about Hippasus is presented with uncertainty. The who is credited caveat means Hippasus may or may not have discovered irrationality. Because of the use of modals before murdered, even if Hippasus had discovered irrationality, the famous associated event his murder may be true, or may be false. Liu s paper serves as a good highlight of the problem with the May Have Camp. Namely, using may have to describe the earliest Greek mathematicians and their mathematical contributions is now out of place in a world that contains new historical research (such as contained in Martinez s book) on these figures. Let me give you an example of this historical research:
On the question of if Hippasus was the first to discover irrationality, Martinez found the tie between Hippasus and irrationality to be the result of the blending [of] two mythical stories about death at sea (2012, p. 23). These two stories come from Iamblichus, who wrote six centuries after Hippasus s lifetime. Iamblichus wrote one story about Hippasus. The contents are essentially: Hippasus was a Pythagorean who first revealed how to inscribe a figure of twelve pentagons into a sphere, and he died at sea for committing impiety (Martinez, 2012, p. 24). Iamblichus wrote another story about the discoverer of irrationality and does not mention it was Hippasus. Its contents are essentially: Someone who first revealed incommensurablity to the unworthy was hated so violently, they say, that he was banished and a tomb constructed for him. Some others say instead this person died at sea as an offender against the gods (Martinez, 2012, p. 24). Then, according to Martinez, in 1892, John Burnet, wrote in Early Greek Philosophy that our tradition says that Hippasos of Metapontium was drowned at sea for revealing this skeleton in the cupboard (Martinez, 2012, p. 21). Burnet attributed his story to Iamblichus, which we can see said no such thing. Therefore, the best existing evidence is that John Burnet mistakenly or purposefully tied separate stories of Pythagoreans dying at sea together, thus tying Hippasus to the discovery of irrationals. On the question of if the discoverer of irrationality was murdered again, there is no evidence this was Hippasus we can see that Iamblichus only wrote that the discoverer may have died at sea, or may have been banished, or may have been banished and then died at sea. Further, died at sea does not necessarily mean drowning. Burnet is the one who embellished with was
drowned at sea, so not only a specific mode of death, but also a passive tense, meaning there could have been an active murderer. But, Burnet could have meant the gods or fate was the active murderer. So then, according to Martinez (2012), it was Morris Kline in 1972, who further embellished was drowned at sea with the new morsel that it was Pythagoreans who threw the discoverer overboard. Now, let us compare Martinez s research on Hippasus with Liu s passage on Hippasus. When Martinez has presented so much information to show how unlikely it is that Hippasus discovered irrationality, is it consistent and sensible for Liu to write (through caveat) that Hippasus may have discovered irrationality? When Martinez has presented so much information to show how unlikely it is that the discoverer of irrationality was murdered and again, how this person was likely not Hippasus is it consistent and sensible for Liu to write that Hippasus may have been murdered? I argued that verbs of certainty were too strong to be used to describe Pythagoras. Then I showed how Liu doesn t use verbs of certainty, and relies on the modal may have, and caveats that perform an equivalent service, to describe an ancient Greek figure (Hippasus). Now, by placing Martinez s research and Liu s passage next to each other for contrast, I am arguing that because of the advent of research such as Martinez s, even the uncertain may have is now too strong to be used to describe Pythagoras and similar ancient Greek figures. If may have means evens probability, as it does for Azar, it is far too strong to describe the possibility of Hippasus discovering irrationality. Do we really believe there is an evens probability Hippasus did so? Even if we weaken may have substantially to the threshold of
something akin to decent shot, it is still far too strong. Do we really believe there is a decent shot that Hippasus discovered irrationality? Again, no. Martinez s research allows us to see Hippasus discovering irrationality as a tiny, tiny possibility far too tiny of a possibility to be consistent with may have. Little to No Evidence Camp: There is LITTLE TO NO EVIDENCE that Pythagoras was a great mathematician, let alone one at all. There is LITTLE TO NO EVIDENCE he made any direct mathematical contributions, let alone great ones such as A, B, C The new reality for writers writing about Hippasus or Pythagoras or their contemporaries will be to say, it is unlikely that Pythagoras rather than Pythagoras may have And if a writer wants to avoid dealing with probabilities and likelihoods that Pythagoras was this, or did that, altogether, they can stick to language about evidence and avoid language about probabilities: They can say the above. But the key takeaway is that those writing about Pythagoras and his contemporaries no longer have safety in the May Have Camp. They must switch camps. I will end now by presenting Martinez s research on the topic of Pythagoras proving the Pythagorean theorem, which is another example of how may have is just not consistent with the tiny probability of this having happened. After Pythagoras died, for 400 years, there s no evidence he found or proved any geometric theorem (Martinez, 2012). Plato never said Pythagoras found any geometric theorem. There is no evidence Aristotle attributed anything in mathematics to Pythagoras. Euclid wrote The Elements in roughly 250 B.C.E., and discusses and provides two proofs of the hypotenuse theorem, but does not mention Pythagoras at all. In 225 B.C.E., Archimedes wrote several comments about the history of geometry, but does not mention any contributions by Pythagoras (Martinez, 2012).
Then, in 45 B.C.E., Cicero writes essentially: Pythagoras found something new in geometry and is said to have immolated an ox for the muses; but I do not believe this (Martinez, 2012). In 15 B.C.E., Vitruvius writes essentially: Pythagoras found that in a 3-4-5 triangle the square on side 5 sums the squares on the others, and allegedly he therefore sacrificed for the Muses (Martinez, 2012). So, Cicero s writing introduces an ox sacrifice, but not the Pythagorean theorem. Vitruvius s writing introduces the Pythagorean theorem but only for the 3-4-5 triangle. Following these two, a number of Greeks between 100-300 C.E., including Apollodurus, Plutarch, Athenaeus, Diogenes, and Porphyry all write that Pythagoras made some type of ox sacrifice (one or many) upon finding a theorem sometimes the hypotenuse theorem and sometimes not (Martinez, 2012). Again, there is no mention of a proof of the theorem. So when was the first mention in history that Pythagoras proved the Pythagorean theorem, and by whom? It was by Galileo in 1632, over 2,000 years after the death of Pythagoras. He essentially wrote: Pythagoras first knew that the square on the hypotenuse equals the squares on the triangle s other sides, and then he proved it and sacrificed a hetacomb [many ox] (Martinez, 2012, p. 11). So, what is the probability that Pythagoras proved the Pythagorean theorem, given the first person to make such a claim was Galileo over 2,000 years later? The probability is too low for may have to be an acceptable answer; there is no more safety in that camp.
References Azar, B. S. (2006). Basic English Grammar (3rd ed.). White Plains, NY: Pearson Longman. Ji, S. (2016). Lecture 4. Pythagoras Theorem and the Pythagoreans [PDF document]. Liu. Hippasus and Evolution [PDF document]. Maor, E. (2007). The Pythagorean Theorem: A 4,000-Year History. Retrieved September 15, 2016, from http://www.luc.edu/facultyauthors/maor_eli.shtml Martińez, A. A. (2012). The Cult of Pythagoras: Math and Myths. Pittsburgh, PA: University of Pittsburgh Press.