Ketuvot 103 ~ The Death of the Editor

The Talmud has been discussing the obligations of heirs to their father’s widow. As an example, it records the instructions that Rabbi Yehudah HaNasi – known to all simply as Rebbi – gave to his sons to honor his widow. This is how he began:"הזהרו בכבוד אמכם -be careful of your mother's honor.”

כתובות קג, א

ת"ר בשעת פטירתו של רבי אמר לבני אני צריך נכנסו בניו אצלו אמר להם הזהרו בכבוד אמכם נר יהא דלוק במקומו שולחן יהא ערוך במקומו מטה תהא מוצעת במקומה

The rabbis taught in a Baraisa: At the time of Rebbi's death he said "I need my sons." They came to him and he said to them: "[After I die] be careful of you mother's honor. Leave a lamp to burn [for me] in its usual place. Let [my usual place at] the table be set, let [my] bed be ready in its usual place...

As the story spills onto daf 104, the Talmud hints at the cause of Rebbi’s demise. He seems to have been suffering from an intestinal disorder, since Rebbi’s maid noted that he needed to use the latrine very often. This was causing him great distress –although apparently the distress was not because he needed to move his bowels so often, but rather that as a result of his condition, he could not wear tefillin. 

כתובות קד, א

הָהוּא יוֹמָא דְּנָח נַפְשֵׁיהּ דְּרַבִּי, גְּזַרוּ רַבָּנַן תַּעֲנִיתָא, וּבְעוֹ רַחֲמֵי, וְאָמְרִי: כֹּל מַאן דְּאָמַר ״נָח נַפְשֵׁיהּ דְּרַבִּי״ — יִדָּקֵר בַּחֶרֶב

It is related that on the day that Rabbi Yehuda HaNasi died, the Sages decreed a fast, and begged for divine mercy so that he would not die. And they said: Anyone who says that Rabbi Yehuda HaNasi has died will be stabbed with a sword.

סְלִיקָא אַמְּתֵיהּ דְּרַבִּי לְאִיגָּרָא, אָמְרָה: עֶלְיוֹנִים מְבַקְּשִׁין אֶת רַבִּי, וְהַתַּחְתּוֹנִים מְבַקְּשִׁין אֶת רַבִּי. יְהִי רָצוֹן שֶׁיָּכוֹפוּ תַּחְתּוֹנִים אֶת הָעֶלְיוֹנִים. כֵּיוָן דַּחֲזַאי כַּמָּה זִימְנֵי דְּעָיֵיל לְבֵית הַכִּסֵּא וְחָלַץ תְּפִילִּין וּמַנַּח לְהוּ וְקָמִצְטַעַר, אֲמַרָה: יְהִי רָצוֹן שֶׁיָּכוֹפוּ עֶלְיוֹנִים אֶת הַתַּחְתּוֹנִים

The maidservant of Rabbi Yehuda HaNasi ascended to the roof and said: The upper realms are requesting the presence of Rabbi Yehuda HaNasi, and the lower realms are requesting the presence of Rabbi Yehuda HaNasi. May it be the will of God that the lower worlds should impose their will upon the upper worlds. However, when she saw how many times he would enter the bathroom and remove his phylacteries, and then exit and put them back on, and how he was suffering with his intestinal disease, she said: May it be the will of God that the upper worlds should impose their will upon the lower worlds.

Rabbi Yehudah HaNassi, Editor Extraordinaire

According to the great scholar of the Talmud David Halivni, the Mishnah came into being 

...as a result of the exigencies of the post-Temple era...towards the second half the of the century with the termination of the oppressive Roman regimens, the Mishnah continued to flourish through the activities of the enormously prestigious R. Judah Hanassi...only to collapse of its own weight soon after R. Judah Hanassi's death.  

As a result, relatively few additions entered the Mishnah; it basically remained much the same as it was when compiled by the editor-anthologist.  This is why the Mishnah is the only classical rabbinic book about whose editor we are relatively certain.  We have no idea who the editors were of any of the other classic rabbinic texts (including the Talmud) but the evidence clearly indicates that R. Judah Hanassi was the editor-anthologist of the Mishnah.  This evidence is based on two sources: the occasional cross reference by R. Yochanan to R. Judah as editor-anthologizer and, above all, the fact that no one who lived after R. Judah Hanassi is mentioned in the Mishnah. 

The Medical History of Rabbi Yehudah HaNassi

Recent scholars have been tempted to diagnose the many illnesses from which Rebbi suffered. In her Hebrew paper The Illnesses of Rabbi Yehudah HaNassi in Light of Modern Medicine, the historian Estée Dvorjetski from the University of Haifa noted that more is known about the ailments of Rebbi than about any other talmudic sage. Some think that Rebbi suffered from painful hemorrhoids, to such a degree that his cries could be heard when he used the latrine ( and as described in Bava Metziah 85a). Rebbi was so distressed by this illness that he ascribed to it a religious meaning, and proclaimed: “The righteous die though intestinal diseases.” But as Divorshki correctly notes, hemorrhoids are not painful to the degree described in the Talmud (– unless complicated by anal fissures). She therefore suggests that Rebbi’s illness - the one from which he died - was an inflammatory bowel disease.

Rebbi suffered from a number of other diseases throughout his life. In Nedarim  we learn that he had episodes of temporary memory loss. He was also afflicted with צמירתא and צפרנא (Bava Metziah 85a). Divorshki the historian notes that some have suggested that צמירתא is kidney stones, perhaps complicated with urinary tract infections. As for צפרנא, (or, in variant forms, צפדנא) Avraham Steinberg from Sha'arei Tzedek Hospital suggests that since this disease was characterized by bleeding from the gums, “it seems reasonable to identify this illness with scurvy.” Julius Preuss had a similar suggestion, one he offered with great certainty: “There can be no doubt that tzafdina refers to stomatitis, perhaps scorbutic stomatitis which also occurs sporadically.” And if these were not enough, Rebbi also had an eye ailment, which his personal physician Shmuel was able to cure, as well as inflammation of his joints, (Yerushalmi Shabbat 16:1) that suggests the illness we call gout. 

A Unifying Diagnosis?

Can a wise clinician put all this together and come up with a single unifying diagnosis that can explain all of Rebbi’s terrible symptoms? In 1978, Ari Shoshan suggested in Korot, The Israel Journal of the History of Medicine and Science, that Rebbi suffered from a psychosomatic disease. However, Divorshki suggests that the rapidly advancing field of genetics can provide a more satisfying solution. She posits that Rebbi had a seronegative spondyloarthritis associated with a specific tissue type called HLA (Human Leukocyte Antigen) B-27. (Don't be afraid. Seronegative means that the condition is not associated with rheumatoid factor, and spondyloarthritis is a group of conditions that causes inflammation of the joints - and other tissues.)  This tissue disorder –a kind of autoimmune disease - is associated with gout (Rebbi had that) and inflammation of the mouth (check) and uveitis – a painful inflammatory eye condition (also that). Perhaps, Divorshki notes, צמירתא was not in fact kidney stones or a urinary infection, but an inflammation of the bladder wall or referred pain from an inflammation of the intestines, caused by the same nasty tissue disorder. For reasons that are still not known, this autoimmune disease can flare up and then, just as mysteriously, become dormant for months or years, which could explain how Rebbi appeared to have been cured.

Schematic ribbon diagram of the HLA-B27 molecule’s peptide-binding cleft with a bound peptide (light blue); the letters N and C indicate, respectively, the amino and carboxy termini of the bound peptide. HLA-B*27:06, one of the two subtypes that seem to have no association with ankylosing spondylitis, and the disease-associated subtype HLA-B*27:04 (from which Rebbi may have been suffering) differ from each other by two residues at positions 114 and 116. From Khan, MA.  Polymorphism of HLA-B27: 105 Subtypes Currently Known.  Current Rheumatology Reports. (2013) 15:362 

We now have identified at least 105 subtypes of HLA-B27, and the list continues to grow.  Today, seronegative spondyloarthitis, of the sort that may have afflicted Rebbi, can often be managed with medications that suppress the immune response. But without these, damage to the host tissues slowly builds until the organ systems start to fail, offering no respite from the painful symptoms of this disease. Perhaps now we are in a position to better understand Rebbi’s dying words, which appear on daf  104a.

“May it be Your will that there will be peace when I rest in eternity.”

Rebbi wanted nothing more than respite from his pain, and his wish was granted: ‘A voice from heaven emerged and said: “He will come with peace, they will rest on their resting places.”

Talmudology Redux ~ Sukkot and Juggling Records

In a discussion of the joyous celebration that would take place in the Temple on the festival of Sukkot, we read this charming passage:

סוכה נג, א

תַּנְיָא: אָמְרוּ עָלָיו עַל רַבָּן שִׁמְעוֹן בֶּן גַּמְלִיאֵל, כְּשֶׁהָיָה שָׂמֵחַ שִׂמְחַת בֵּית הַשּׁוֹאֵבָה, הָיָה נוֹטֵל שְׁמֹנֶה אֲבוּקוֹת שֶׁל אוֹר, וְזוֹרֵק אַחַת וְנוֹטֵל אַחַת וְאֵין נוֹגְעוֹת זוֹ בָּזוֹ

They said about Rabban Shimon ben Gamliel that when he would rejoice at the Celebration of the Place of the Drawing of the Water, he would take eight flaming torches and toss one and catch another, juggling them, and, though all were in the air at the same time, they would not touch each other.

This might sound like a just another neat trick, but as a very amateur juggler, I can assure you that it is much more that that. It is almost impossibly difficult. Rabbi Shimon ben Gamliel must have spent hours and hours perfecting this juggling ability, but difficult as it most certainly is, it is entirely achievable.

world records for juggling clubs

To get a sense of how difficult this sort of thing is, let’s start with juggling “only” seven clubs. Here is American juggler Anthony Gatto juggling them for a world record four minutes (and 24 seconds). No need to watch the whole thing, though it is very magical.

Now here is Gatto juggling with eight clubs and setting a then world record in 2006. Watch the twelve second clip, and then imagine doing that with clubs that were on fire.

Finally, here is the current world record for juggling clubs. It was set by the Norwegian Eivind Dragsjø in 2016. He managed to juggle nine clubs for eleven catches.

Now, with knives

This page of Talmud tells us of another rabbinic juggler by the name of Levi, actually out-juggled Rabban Shimon ben Gamliel:

לֵוִי הֲוָה מְטַיֵּיל קַמֵּיהּ דְּרַבִּי בְּתַמְנֵי סַכִּינֵי

Levi would walk before Rabbi Yehuda HaNasi juggling with eight knives.

Juggling knives is much harder than juggling balls or clubs, because, well, they are knives and have a sharp end. The current world record for juggling knives appears to be six. No-one today has yet juggled eight knives, or even seven for that matter. Oh, and by the way, if you are thinking of attempting to set this record, here is the small print:

WARNING: This record can be extremely dangerous. Please do not attempt this record unless you are above the age of 18 and trained as a professional juggler. We will not accept submissions in this category from minors.

A More Gentle Approach

Elsewhere in the Talmud we read of another sage who juggled, though he was criticized for his skill:

כתובות יז, א

אמרו עליו על רבי יהודה בר אילעאי שהיה נוטל בד של הדס ומרקד לפני הכלה ואומר כלה נאה וחסודה רב שמואל בר רב יצחק מרקד אתלת א"ר זירא קא מכסיף לן סבא כי נח נפשיה איפסיק עמודא דנורא בין דידיה לכולי עלמא וגמירי דלא אפסיק עמודא דנורא אלא אי לחד בדרא אי לתרי בדרא

The Sages said about Rabbi Yehuda bar Elai that he would take a myrtle branch and dance before the bride, and say: A fair and attractive bride. Rav Shmuel bar Rav Yitzchak would base his dance on three myrtle branches that he would juggle. Rabbi Zeira said: The old man is humiliating us, as through his conduct he is demeaning the Torah and the Torah scholars…

רש׳י שם

אתלת - שלש בדין זורק אחת ומקבל אחת

Three - he would take three branches, and toss them in the air and catch them

Another Sage WHo Juggled

So although Rabbi Yehudah juggled with myrtle branches, which are far safer than knives, he was criticized for conduct unbecoming a Talmudic sage. But that seems a bit harsh. In any event, we may no longer have juggling sages, but we do have juggling mathematicians, who are sages of a different order. The foremost of these was the late Ronald Graham (1935-2020) who was a quite brilliant mathematician; in 2003, he was awarded the American Mathematical Society's annual Leroy P. Steele Prize for Lifetime Achievement. Graham wrote six books and some 400 mathematical papers, many with the famous Hungarian mathematician Paul Erdos. But Graham was also a juggler; he began when he was 15, and was able to juggle six balls. In 1973 he was elected President of the International Jugglers’ Association. It should come as no surprise that he also wrote an important paper on the mathematics behind juggling drops and descents.

The acclaimed mathematician Ronald Graham juggling a four-ball fountain (1986).

Rabban Shimon ben Gamliel, Levi, and Rabbi Yehudah demonstrate that the rabbis of the Talmud had hobbies, that they worked to perfect them, and they became rather good at them. Ronald Graham was not a trailblazer; instead, and unbeknown to him, he was following an esteemed Talmudic tradition.

Wishing you all a joyous Sukkot, whether or not you can juggle.

Talmudology Redux ~ Could the Vilna Gaon Have Stayed Awake Over Sukkot?

The Vilna Gaon, (lit. The Genius of Vilna) lived from 1720-1797, and next week, the 19th of Tishrei, is the Hebrew anniversary of his death. He was one of the leading talmudic and halakhic authorities of his time. His works are widely studied to this day, and his legal opinions are often cited. Today’s question is: did the Vilna Gaon stay awake for several consecutive days one Sukkot? Here is the background:

The Vilna Gaon.

In January 1788 the Vilna Gaon was involved in the kidnapping of a young Jewish man who had converted to Christianity. For his role in the kidnapping Rabbi Eliyahu (the given name of the Gaon) was arrested and held for over a month.  The case was later tried, and on September 15, 1789 (sic) the Gaon of Vilna, together with others involved in the kidnapping, was sentenced to twelve weeks in jail.  Although it is unclear how long he was imprisoned, he was there over Sukkot, and the Lithuanian authorities were hardly in the practice of providing imprisoned Jews with a sukkah.  But since one is not permitted to sleep outside of the sukkah, what was the Gaon to do?  Simple.  He’d stay awake, and by doing so he would not transgress the prohibition of sleeping outside the sukkah.  Here’s how the episode is described in the work Tosefet Ma’aseh Rav published in 1892. 

Our Leader, teacher and Rabbi may he rest in peace, when he was imprisoned on Sukkot, tried with all his strength, and walked from one place to another, and held his eyelids open, and made an extraordinary effort not to sleep outside the sukkah – not even a brief nap – until the authorities released him to a sukkah.

We don’t know for how many nights the sixty-nine year-old Rabbi Eliyahu stayed awake, but is such a claim even plausible? As it turns out, it is.

The World Record for Staying Awake

The world record for staying awake is an amazing eleven days. Eleven days – that’s 264 hours (and 24 minutes to be precise) without sleep. It was set in 1965 by Randy Gardner, who was then seventeen years old. Gardner seems to have suffered little if any harm by his marathon period of sleep deprivation. But don’t try and beat the record. The Guinness Book of Records no longer has an entry for staying awake – because it is considered too dangerous an ordeal to undertake. (You can hear a review of sleep deprivation stunts in general and a wonderful interview with Gardner himself here.)

The Health Risks of Sleep Deprivation

What are the health risks of prolonged sleep deprivation? A 1970 study of four volunteers who stayed awake for 205 hours (that’s eight and a half days!) noted some differences in how the subjects slept once they were allowed to do so, but follow-up testing of the group conducted 6-9 months after the sleep deprivation showed that their sleep patterns were similar to the pre-deprivation recordings.

“Sleep deprivation reduces learning, impairs performance in cognitive tests, prolongs reaction time, and is a common cause of seizures. In the most extreme case, continuous sleep deprivation kills rodents and flies within a period of days to weeks. In humans, fatal familial or sporadic insomnia is a progressively worsening state of sleeplessness that leads to dementia and death within months or years. ”

— Lulu Xie et al. Sleep Drives Metabolite Clearance from the Adult Brain. Science 2013. 342.317

Although Randy Gardner and these four volunteers seem to have suffered no long-term health consequences of staying awake for over a week, scientists have long noted that sleep deprivation is rather bad for the body. Or to be more precise, the bodies of unfortunate laboratory rats who are not allowed to sleep. In these animals, prolonged sleep deprivation causes the immune system to malfunction. This results in infection and eventual lethal septicemia. The physiologist who kept these rats awake noted that there are “far-reaching physical implications resulting from alterations in immune status [which] may explain why sleep deprivation effects are risk factors for disease and yet are not well defined or specifically localized.” In other words, sleep deprivation makes rats really sick, but we don’t know how, or why…

One possible explanation was suggested in 2013 by a group from the University of Rochester Medical Center. They demonstrated that during sleep, the space between the cells of the brain (the interstitial space) increased by up to 60%, allowing toxic metabolites to be cleared. This raises the question of whether the brain sleeps in order to expel these toxic chemicals, or rather it is the chemicals themselves that drive the brain to switch into a sleep state.

The extracellular (interstitial) space in the cortex of the mouse brain, through which cerebral spinal fluid moves, increases from 14% in the awake animal to 23% in the sleeping animal, an increase that allows the faster clearance of metabolic waste products and toxins. From Suzana Herculano-Houzel. Sleep it out. Science 2013: 342; 316

Not having any sleep is bad for your health - but so too is going without enough sleep. Chronic restriction of sleep to six hours or less per night can produce cognitive performance deficits equivalent to up to two nights of total sleep deprivation. So be sure to get a full night's rest.

Hard, But Not Impossible

The world record set by Randy Gardner has implications for a talmudic decision. In the tractate Shavuot (26b) Rabbi Yochanan ruled that since it was impossible to stay awake for more than three days, any vow to do so is considered to have been a vow made in vain – and punishment follows swiftly.  Here is how the ArtScroll Talmud explains this ruling, (based on the explanation of the Ran).

Since it is impossible for a person to go without sleep for three days, the man has uttered a vain oath. Hence, he receives lashes for violating the prohibition (Exodus 20:7): לא תשא את שם ה׳ אלוקיך לשוה, You shall not take the Name of Hashem, your God, in vain. And since the oath -being impossible to fulfill -has no validity, he is not bound by it at all and may sleep immediately.

Maimonides codified this law and also assumed that it is impossible to stay awake for three consecutive days. 

רמב"ם הלכות שבועות פרק ה הלכה כ

נשבע שלא יישן שלשת ימים, או שלא יאכל כלום שבעת ימים וכיוצא בזה שהיא שבועת שוא, אין אומרין יעור זה עד שיצטער ויצום עד שיצטער ולא יהיה בו כח לסבול ואח"כ יאכל או יישן אלא מלקין אותו מיד משום שבועת שוא ויישן ויאכל בכל עת שירצה

If a person swore that he would not sleep for three days, or would not eat anything for seven days, or something similar, this is considered a false oath. We do not tell the person to stay awake until it is impossible to do so, or fast for as long as possible until the discomfort is too great to bear, and then allow him to eat or sleep. Rather he is punished with lashes immediately for making a false oath, and is then allowed to sleep or eat as much as he wants…

Based on Randy Gardner’s feat, Rabbi Yochanan was incorrect when he ruled that it was impossible to stay awake for three days. It's certainly not impossible, but that hardly means it's a good idea to try. 

It also means that the story of the Gaon of Vilna’s sleepless nights in that cold prison might indeed have occurred. Still it is best not to disrupt your usual sleep patterns. Perhaps that is why Rabbi Chaninah ben Chachina'i in Masechet Avot  (3:4) taught that one who stays awake at night "will forfeit with his life."  Now that's a warning to heed.

רבי חנינא בן חכינאי אומר הנעור בלילה ... הרי זה מתחייב בנפשו

HAPPY SUKKOT FROM TALMUDOLOGY

Ketuvot 93a ~ Game Theory and the Principle of Contested Sums

We are currently studying a series of cases in co-wives claim the money from their ketuvah when the husband dies. It gets complicated.

משנה, כתובות צג, א

  מי שהיה נשוי שלש נשים ומת כתובתה של זו מנה ושל זו מאתים ושל זו שלש מאות ואין שם אלא מנה חולקין בשוה

היו שם מאתים של מנה נוטלת חמשים של מאתים ושל שלש מאות שלשה שלשה של זהב

היו שם שלש מאות של מנה נוטלת חמשים ושל מאתים מנה ושל שלש מאות ששה של זהב

If a man who was married to three wives died, and the kethubah of one was a maneh (one hundred zuz), of the other two hundred zuz, and of the third three hundred zuz, and the estate was worth only one maneh (one hundred zuz), they divide it equally. 

If the estate was worth two hundred zuz, the claimant with the kesuva of the maneh receives fifty zuz,  while the and the claimants of the two hundred and the three hundred zuz each receive three gold denarii (worth seventy-five zuz).

If the estate was worth three hundred zuz, the claimant of the maneh receives fifty zuz, the claimant of the two hundred zuz receives a maneh (one hundred zuz) and the the claimant of the three hundred zuz receives six gold denarii (worth one hundred and fifty zuz)…

In these cases, each surviving wife claims not all, but only part of the deceased husband’s estate. We find a similar contested case in the opening pages of Bava Metziah. Here, two people claim ownership of a garment; one claims she owns all of it, and the other that she owns 50%:

בבא מציעא ב,ב

שנים אוחזין בטלית ... זה אומר כולה שלי וזה אומר חציה שלי ...זה נוטל שלשה חלקים וזה נוטל רביע

Two hold a garment; ... one claims it all, the other claims half. ... Then the one is awarded 3⁄4, the other 1⁄4.

In this case, the Mishnah rules that each swears under oath, and then the garment is divided with 3/4 awarded to one claimant and 1/4 to the other.

Rashi and the Principal of Contested Sums

In his explanation of the case in Bava Metziah, Rashi notes that the claimant to half the garment concedes that half belongs to the other claimant, so that the dispute revolves solely around the second half. Consequently, each of them receives half of this disputed half - or a quarter each.:

זה אומר חציה שלי. מודה הוא שהחצי של חבירו ואין דנין אלא על חציה הלכך זה האומר כולה שלי ישבע כו' כמשפט הראשון מה שהן דנין עליו נשבעין שניהם שאין לכל אחד בו פחות מחציו ונוטל כל אחד חציו

Now of course this is only one way that the garment could be divided between the two claimants. For example, it could be divided in proportion to the two claims, (2/3-1/3), or even split evenly (1/2-1/2).  But instead, and as Rashi explained, the Mishnah ruled using the principal of contested sums. Which is where Robert Aumann comes in.

Game Theory from Israel's Nobel Prize Winner

We have met Robert Aumann before, when we reviewed Israel's glorious winners of the Nobel Prize. For those who need reminding, Aumanm, from the Hebrew University, won the 2005 Nobel Prize in Economics. His work was on conflict, cooperation, and game theory (yes, the same kind of game theory made famous by the late John Nash, portrayed in A Beautiful Mind). Aumann worked on the dynamics of arms control negotiations, and developed a theory of repeated games in which one party has incomplete information.  The Royal Swedish Academy of Sciences noted that this theory is now "the common framework for analysis of long-run cooperation in the social science." The kippah-wearing professor opened his speech at the Nobel Prize banquet with the following words (which were met with cries of  אמן from some members of the audience): 

ברוך אתה ה׳ אלוקנו מלך העולם הטוב והמיטב

If you haven't already seen it, take the time to watch the four-minute video of his acceptance speech. It should be required viewing for every Jewish high school student (and their teachers).

Where was I? Oh, yes. Contested sums.  In 1985, twenty years before receiving his Nobel Prize, Aumann described the theoretic underpinnings of today's Mishnah, as part of a larger discussion about bankruptcy.  His paper, published in the Journal of Economic Theory, is heavy on mathematical notations and light on explanations for non-mathematicians (like me).  Fortunately he later published a paper that is much easier to read and which covers the same material.  The second paper appeared in the Research Bulletin Series on Jewish Law and Economics, published by Bar-Ilan University in June 2002.  "Half the garment" wrote the professor, "is not contested: There is general agreement that it belongs to the person who claimed it all. Hence, first of all, that half is given to him. The other half, which is claimed by both, is then divided equally between the claimants, each receiving one-quarter of the garment." Here is how Aumann visualizes it:

There is another example of this from the Tosefta, a supplement to the Mishnah and contemporary with it. In this new case, one person claims the entire garment, and one claims only one third of it. In this case, the first person gets 5/6 and the second gets 1/6.  

Aumann calls this principal the "Contested Garment Consistent." And this this principle is found in other contested divisions, like our case in Ketuvot 93a, in which a man dies, leaving debts totaling more than his estate.

Aumann likes to think of it this way:

A Nobel Explanation of today’s daf

Here is how he explains what is going on in today’s page of Talmud.

We’ll call the creditor with the 100-dinar claim “Ketura,” the one with the 200, “Hagar” and the one with the 300, “Sara.” Let’s assume, to begin with, that the estate is 200. As per [the table above], Ketura gets 50 and Hagar 75 – together 125. On the principle of equal division of the contested sum, the 125 gotten by Hagar and Ketura together should be divided between them in keeping with this principle. ... In other words, the Mishna’s distribution reflects a division of the sum that Hagar and Ketura receive together according to the principle of equal division of the contested sum...
The division of the estate among the three creditors is such that any two of them divide the sum they together receive, according to the principle of equal division of the contested sum. This precisely is the method of division laid down in the Mishna in Bava Metzia that deals with the contested garment. 

There's a lot more to the paper, including an interesting proof of the principal using fluids poured into cups of different sizes.  But I prefer to focus on another aspect of the paper.  Prof. Aumann notes that, in addition to its own internal logic, the underlying principle of contested sums fits in well with other talmudic passages. 

The reader may ask, isn’t it presumptuous for us to think that we succeeded in unraveling the mysteries of this Talmudic passage, when so many generations of scholars before us failed? To this, gentle reader, we respond that the scholars who studied and wrote about this passage over the course of almost two millennia were indeed much wiser and more learned than we. But we brought to bear a tool that was not available to them: the modern mathematical theory of games.

The actual sequence of events was that we first discovered that the Mishnaic divisions are implicit in certain sophisticated formulas of modern game theory. Not believing that the sages of the Talmud could possibly have been aware of these complex mathematical tools, we sought, and eventually found, a conceptual basis for these tools: the principle of consistency. Of this, the sages could have been, and presumably were, aware. It in itself is sufficient to yield the Mishnaic divisions; and it is this principle that we describe below, bypassing the intermediate step – the game theory.

It’s like “Alice in Wonderland.” The game theory provides the key to the garden, which Alice had such great difficulty in obtaining. Once in the garden, though, Alice can discard the key; the garden can be enjoyed without it.

What a wonderful analogy. Game theory is the key to entering the garden, a key which might not have been available to earlier generations who learned the Talmud. How lucky we are.  

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Want more on Today’s Daf? Click here to watch Talmudology reader Dr Shalom Kelman explain a (Non) Game Theory approach.

Shabbat Shalom from Talmudology