Some Examples of Mathematical Analysis Applied to Talmud Study1 Jonathan Rosenberg As a motto for this article I would like to invoke the last from the 3rd Chapter of One can translate this roughly as follows Mishnah Avot Chapter 3 Mishnah 18 Rabbi Eliezer the son of Chisma says and while difficult tractates involving complicated quantitative issues are an integral part of the Astronomy and geometry2 are the embellishments3 of wisdom Thus this speaks both of the value of mathematics in and of itself and of the value of applying mathematical analysis to Talmud study I certainly don t want to claim that the latter should be the normal way to study Talmud as in fact it only applies to a few isolated But if we are to take the idea of seriously then we should entertain the idea that mathematics even serious mathematics might sometimes have something useful to say about Talmud study So I would like to consider three examples Example 1 Probability and Family Planning 1 Based on a talk given at KMS on January 20 2001 Everyone agrees that the term comes from the Greek but some say that it corresponds to our word geometry and others say that it comes from the same root gramma meaning letter as our word grammar The latter is more likely when the term refers to the numerical value of letters but here geometry literally in Greek earth measurement is a better match for astronomy 3 This is a gastronomic metaphor The term means a side dish or dessert Such a course is not central to the meal but it often makes the difference between a humdrum meal and a memorable one so the shouldn t be ignored 2 Mishnah Yevamot Chapter 6 Mishnah 7 A man should not abstain from having children unless he already has children How many are needed to fulfill one s obligation Beit Shammai say two males and Beit Hillel say a male and a female as it is said He created them male and female Let s see what the consequences of this Mishnah are Suppose a couple starts to have children and agrees they will continue until they have fulfilled their obligation of What are the probabilities of the various family sizes at the time they complete their obligation Let s assume the chance of a boy each time is and that the sexes of the children are independent events Possible sequences of children according to in the First we assume they follow Beit Hillel Note that whatever the sex of the first child the couple according to just continue till they get a child of the opposite sex So the possibilities are BG probability 1 4 GB probability 1 4 BBG probability 1 8 GGB probability 1 8 BBBG probability 1 16 GGGB probability 1 16 etc Thus if X is the number of children P X 2 1 4 1 4 1 2 P X 3 1 8 1 8 1 4 etc Now an important concept in probability theory is that of expected value or average value where we weight events by their probabilities So the expected number of children is the sum of various possible numbers of children each weighted by the corresponding probability i e n 1 3 Note that as this model treats girls n n 1 2 and boys completely equally and both are equally likely the expected number of girls per family is 3 2 and the expected number of boys per family is 3 2 E X 2 3 1 4 4 1 8 Possible sequences of children according to in the Next suppose the couple follow Beit Shammai This time the couple must continue until they have two boys regardless of the number of girls So the possible sequences are BB probability 1 4 GBB probability 1 8 BGB probability 1 8 GGBB probability 1 16 GBGB probability 1 16 BGGB probability 1 16 etc Thus if X is the number of children P X 2 1 4 P X 3 1 8 1 8 1 4 P X 4 1 16 1 16 1 16 3 16 etc So expected number of children is the sum of various possible numbers of children each weighted by the corresponding probability i e n n 1 4 Note that since the couple 2 n 1 n 1 always ends up with two boys the expected number of boys is 2 and the expected number of girls is E X 2 2 An interesting consequence is that the misogynist opinion of is the one that results in the greatest number of girls an average of two per family compared with an average of one and a half per family for E X 2 1 4 3 2 8 4 3 16 Now let s take a look at some of the Gemara on our Mishnah Gemara Yevamot 62a It was taught in a says Beit Shammai say two males and two females and Beit Hillel say a male and a female says what s the reasoning of according to Beit Shammai It s because it is written she continued to give birth to his brother Abel Abel and his sister Cain and his sister It was taught in another says Beit Shammai say a male and a female and Beit Hillel say either a male or a female says what s the reasoning of according to Beit Hillel It s because it is written He didn t create it the world for naught to remain barren Note that there are two different versions of the Beit Hillel Beit Shammai controversy quoted here in the name of and neither one agrees with the version in the Mishnah But now that we ve done our calculation there is an unexpected payoff we can reconcile the two statements of that seem to contradict both each other and the with the formulation in the One can argue that the first is talking about expected value so that is why says Beit Shammai say two boys and two girls and why he says Beit Hillel say a boy and a girl It s natural to round down from the actual expected values of 1 5 and 1 5 since an individual family can t have fractional children Furthermore the prooftext quoted in support of Beit Shammai is consistent with our interpretation since the sisters of Cain and Abel are not mentioned explicitly in Genesis they just come along for the ride since on average waiting for a second boy entails having two girls also On the other hand since the whole purpose of having children is perpetuation of the species which requires a boy and girl in the next generation to replace their parents perhaps it s taken for granted that everyone will have at least a boy and a girl 4 and the second is talking about the excess of the expected value over the norm of one girl one boy This agrees precisely with saying that Beit Shammai say a boy and a girl and that Beit Hillel say a boy or a girl Next let s examine the …