A project of Bar-Ilan University's Faculty of Jewish Studies, Paul and Helene Shulman Basic Jewish Studies Center, and the Office of the Campus Rabbi. Sponsored by Dr. Ruth Borchard of the Shoresh Charitable Fund (SCF). Published with assistance of the President's Fund for Torah and Science. Permission granted to reprint with appropriate credit.
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Parashat Behar-Behukotai 5758/1998
"When a few follow the Torah it is not the same
as when many follow the Torah"
Dr. Alex Klein
Department of Mathematics and Computer Science
Leviticus 26:8 reads: "Five of you shall give chase to a hundred, and a hundred of you shall give chase to ten thousand." Rashi asks, "Is this indeed the way the arithmetic goes? Should it not have said a hundred of you will give chase to two thousand? Note, however, that when a few follow the Torah it is not the same as when many follow the Torah."
Rashi observes the strange fact that the ratio of Israelites to their enemies who flee from them is not constant. In the first part of the verse it is 1:20, and in the second part it reaches 1:100. What accounts for this? Rashi explains that one must also take into account the absolute number of children of Israel who follow the Torah. As their numbers increase, so does the ratio, i.e., so does the might of each and every one (on the average) in repulsing the enemies of Israel.
This notion provides the key to understanding an enigmatic passage in Genesis: Abraham's supplication for Sodom and the other cities of the Plain (Gen. 18:23-33). Abraham begins by requesting that the merits of fifty righteous suffice to save five cities. Subsequently he asks that the merits of forty, then thirty, then twenty, and finally ten suffice. Rashi explains that perhaps forty righteous people would be found there, and four cities would be saved; or thirty, and three cities; or twenty, and two cities; or ten would save one. Nahmanides expresses surprise at this interpretation (Gen. 18:24): "If so, what is the point of Abraham's prayer and supplication, each time entreating the Lord, 'Let not my Lord be angry if I go on,' and 'I venture again to speak to my Lord'...? It stands to reason that forty would save four cities, thirty and twenty likewise, according to the same calculations, just as fifty would save five cities."
Nahmanides offers the following explanation: "Many righteous merit greater deliverance than fewer righteous, who might save just a few. As it is said: when a few follow the Torah it is not the same as when many follow the Torah." According to this general rule of the Sages, set forth above, the possibility that all five cities could be delivered on the merits of the many, i.e., fifty people, does not necessarily imply that a single city could be delivered by as few as ten. It is not merely a question of ratios; rather, one must also take into account the absolute number of righteous (cf. Nehama Leibowitz, Studies in Leviticus, Parshat Be-Hukotai).
It still remains to clarify why the power of many who follow the Torah should be greater. The reason apparently lies in the fact that society is more than a group of individuals; reciprocal influences in the realm of faith and morality are constantly at work among the people who comprise the society. Therefore, the more people in the society, the greater the impact exerted on each and every one, for better of for worse.
Many psychological and sociological explanations have been offered for this. I would like to provide an explanation using simple mathematical tools. Let us compute quantitatively the total possible relations between individuals in a society numbering n individuals. We shall call this number k. Using the general combinatoric formual for computing the number of possible combinations of any 2 out of n items, namely k = n * (n - 1) / 2, we obtain (for n = 2 .. 10):
n 2 3 4 5 6 7 8 9 10
k-> 1 3 6 10 15 21 28 36 45