initial speeds of all spins in its causal future unchanged, that would require an extremely delicate operation – arguably a much ‘larger miracle’ than the simple introduction of a counterfactual sneeze. However, there is another way to understand Abrams’s argument here that avoids this difficulty. Consider a typical counterfactual intervention that produces changes in the initial speeds of very many spins that are big enough to move them from one pocket to another. Again imagine the spin-speed line divided up into suitably small pockets, and this time imagine the pockets to be sequentially numbered. Presumably, there is no reason to expect that whether the speed of a given spin gets moved to a different pocket or not, or which direction it gets moved in, or how many ‘pocket-lengths’ it moves, to be correlated with whether it is actually in an even-numbered or odd-numbered pocket. (If I were to sneeze, that would shake up all the spins randomly.) So, on average, we should expect the pockets to lose individual spins to their left-side neighbors and to their right-side neighbors in about equal numbers, and to expect the neighboring pockets to lose individual spins to their left- and right-side neighbors in about the same numbers. (This is simply a point about combinatorics: By far, most of the possible ways of rearranging a large number of spins by moving them from their actual pocket to an adjacent one will be as just described – just as most of the possible ways of writing a sequence of 10,000 1s and 0s have about equal numbers of 1s and 0s.) Similarly, on average, we should expect the pockets to lose individual spins to their neighbors two pockets down on either side at about the same rate, and at about the same rates as the other very nearby pockets do. And so on. So on average, the number of spins located within each pocket should change very little. Therefore, if the actual frequency distribution is approximately macroperiodic, we should expect that typical counterfactual distributions resulting from small interventions will also be approximately macroperiodic. There are possible counterfactual disturbances which would make the distribution become For example, a counterfactual bump could have the effect of moving vast numbers of spins whose initial speeds are actually located in odd-numbered pockets into even-numbered pockets while moving few or no spins from
initial speeds of all spins in its causal future unchanged, that would require an extremely delicate operation – arguably a much ‘larger miracle’ than the simple introduction of a counterfactual sneeze. However, there is another way to understand Abrams’s argument here that avoids this difficulty. Consider a typical counterfactual intervention that produces changes in the initial speeds of very many spins that are big enough to move them from one pocket to another. Again imagine the spin-speed line divided up into suitably small pockets, and this time imagine the pockets to be sequentially numbered. Presumably, there is no reason to expect that whether the speed of a given spin gets moved to a different pocket or not, or which direction it gets moved in, or how many ‘pocket-lengths’ it moves, to be correlated with whether it is actually in an even-numbered or odd-numbered pocket. (If I were to sneeze, that would shake up all the spins randomly.) So, on average, we should expect the pockets to lose individual spins to their left-side neighbors and to their right-side neighbors in about equal numbers, and to expect the neighboring pockets to lose individual spins to their left- and right-side neighbors in about the same numbers. (This is simply a point about combinatorics: By far, most of the possible ways of rearranging a large number of spins by moving them from their actual pocket to an adjacent one will be as just described – just as most of the possible ways of writing a sequence of 10,000 1s and 0s have about equal numbers of 1s and 0s.) Similarly, on average, we should expect the pockets to lose individual spins to their neighbors two pockets down on either side at about the same rate, and at about the same rates as the other very nearby pockets do. And so on. So on average, the number of spins located within each pocket should change very little. Therefore, if the actual frequency distribution is approximately macroperiodic, we should expect that typical counterfactual distributions resulting from small interventions will also be approximately macroperiodic. There are possible counterfactual disturbances which would make the distribution become For example, a counterfactual bump could have the effect of moving vast numbers of spins whose initial speeds are actually located in odd-numbered pockets into even-numbered pockets while moving few or no spins from
VonKriesPaper-final.docx