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
Martínez, A. A. (2012). The Cult of Pythagoras: Math and Myths. Pittsburgh, PA: University of
Pittsburgh Press.