two Jews, three opinions

The annual AAC&U Diversity, Equity, and Student Success conference occurred in Pittsburgh this year at the Omni William Penn. After lunch last Friday, during the discussion on the Tree of Life tragedy, one of the rabbis said if you speak with two Jews, you get three opinions.

I retreated from the emotionally laden conversation to the peace of Diophantine mathematics, and asked myself: what is the general case? With n Jews, how many opinions O(n) will be present? I started to consider possible functions that satisfy the following conditions:

  1. O(2) = 3
  2. O(n) is a counting number for all counting numbers n
  3. O(n) is a polynomial function
  4. O(n) should admit of some reasonable answers in the special cases when n = 0 and n = 1 

O(n) = n + 1 provides the entertaining result that one Jew, being alone, will nevertheless harbor two opinions. When n = 0, the function indicates is there is still one opinion, which could be interpreted as that of God. While this null case is interesting, the interpretation of cases when n > 1 is increasingly dull. The number of opinions is always simply one more than the number of Jews present.

O(n) = 2n – 1 is slightly more interesting with its the steeper slope. As n becomes large, the number of opinions approaches double the number of interlocutors, suggesting that Jews are able to entertain two sides of an argument that each is individually interpreting. However, when n = 0 (the case of “God alone”), the number of opinions is negative, which is not a meaningful result.

Other linear polynomials, such as 3n – 3, yield nonsensically negative results for n = 1 (the individual case) or don’t provide integer solutions across the domain of counting numbers, such as 1/2 n + 2. Likewise, higher-order polynomials such as n^2 – 1, n^3 – 5, and n^4 -13 fail to give meaningful results when n = 1.

O(n) = 3 is the trivial case. This function satisfies all four of the given conditions. However, it doesn’t pass the “sniff test” –I would expect a group of Jews to hold more than 3 opinions among each other.

O(n) = n^2 – n + 1 has the lovely quality of growing faster than the other three viable options. As n approaches infinity, the number of opinions approaches the square of the number of Jews. To put it another way, the number of interactions increases as the number of pairwise interactions between n argumentative, intellectual, inquisitive people: n (n -1). The constant at the end could represent God’s opinion.

I leave it as an exercise to the reader to prove that other polynomial functions besides these four cannot fulfill the conditions above. As for me, I favor the last of these possibilities.

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