Chof Iyar, 5774
The "Trolley Problem" is a well known scenario that has been widely debated. The case, briefly, is as follows: a trolley cart is traveling down a track towards a group of five people-about to run them over. You are able to divert the cart onto a different track by pulling a lever, thereby saving these five lives. However, on this second track is one person who will be killed if you divert the cart.
Do you pull the lever and save the five? How would Halacha deal with this issue?
The Rambam (Hilchos Yesodei HaTorah 5:5) deals with a similar case (based on the Yerushalmi, Terumos, end of chapter 8). He writes that in a case where gentiles demand one Jew from a certain city be handed over to be executed and if not the entire city will be destroyed, it is nevertheless forbidden to give up a single Jewish life, even though doing so would save the entire city. Seemingly, the same rationale could be applied to our case: It would be forbidden to pull the lever, as it would enable one to "give up" that one person, as specified in the Rambam's case.
The Chazon Ish (Sanhedrin, 25) cites a case perfectly analogous to our trolley case. He writes that if someone sees a projectile sailing through the air towards a group of people, and he is able to intercept this projectile, but in doing so he would have to divert it towards a single person-what should he do? This case is an exact match to our "Trolley Problem."
The Chazon Ish suggests that this case may be different from the case of the Rambam, as in this case you are directly saving someone with your action. Therefore, this act may be considered an "act of saving," even if it leads to an individual death. In the case of the Rambam, the act of handing someone over to be killed is an act of killing. Yes, the rest of the town would be saved, but it's not your act directly which saves them; it is merely an outcome of your act of killing. Therefore, suggests the Chazon Ish, this case is not analogous to the case of the Rambam. Here, it would be permitted to divert the projectile (or in our case, pull the lever), since it would be considered an "act of saving" [five lives] and we generally try to minimize [Jewish] death.
The Rebbe, however, (Reshima #123) seems to understand the Rambam as we originally thought, and does not take into account the suggestion of the Chazon Ish. He writes-quoting the Rambam's din-that we see from here that Torah does not value multiple lives over a single life. He understood the Rambam to be setting out a basic premise, and therefore, even in our scenario, the Rambam's din would apply and one would be forbidden to pull the lever. The Rebbe goes on to explain the rationale behind this law, saying that since a Jew's soul is part of the infinite Hashem, two souls are no more an expression of Him than is one soul. Infinite plus infinite is still infinite.