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Infinite aggregation: expanded addition

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How might we extend aggregative moral theories to compare infinite worlds? In particular, how might we extend them to compare worlds with infinite spatial volume, infinite temporal duration, and infinitely many morally valuable phenomena? When doing so, we face various impossibility results from the existing literature. For instance, the view we adopt can endorse the claim that (1) worlds are made better if we increase the value in every region of space and time, or (2) that they are made better if we increase the value obtained by every person. But they cannot endorse both claims, so we must choose. In this paper I show that, if we choose the latter, our view will face serious problems such as generating incomparability in many realistic cases. Opting instead to endorse the first claim, I articulate and defend a spatiotemporal, expansionist view of infinite aggregation. Spatiotemporal views such as this do face some difficulties, but I show that these can be overcome. With modification, they can provide plausible comparisons in cases that we care about most.

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  1. We need not agree on what composes value—it may be pleasure, happiness, a plurality of things, or any other quantity associated with all tokens of some physical phenomena.

  2. The set of such token events will be countably infinite. For any given phenomenon, I’ll assume, each token must occupy some (exclusive) non-zero volume of spacetime. For illustration, a human brain to experience a given quantity of pleasure, it requires some non-zero spatial volume and some non-zero, finite duration. So we can only fit a countably infinite number of those token events into the world.

  3. Physicists are hesitant to speak of infinite quantities (exceptions include Garriga and Vilenkin 2001; Gott 2008), as these can never be empirically confirmed. Nonetheless, this conclusion is entailed by both of the above views.

  4. The latter two options won’t help with the idealized example considered above, but they do help with cases in which we change the identities of many future persons. See Sect. 5.3.

  5. Following others, I pursue an axiological solution rather than a normative one—one which revises how we compare worlds, rather than directly revise how we judge the permissibility of actions (whether objectively or subjectively). This is for three reasons. The first: brevity. The second: for all minimally aggregative views, the problem arises from having an axiology which doesn’t discriminate enough; with an axiological solution, we’ll automatically have a normative solution. And the third: these views are united by their axiology, but differ in all else, so altering the axiology will provide the most general solution.

    There is one proposed solution which directly modifies our (subjective) normative principle rather than axiology, given by Arntzenius (2014). I argue elsewhere that this solution is untenable (Wilkinson n.d.(c)).

  6. When discussing expansionism, Askell (2018, pp. 71–87) also focuses on the spatiotemporal version. I agree that this is the most compelling version, as I argue in Sect. 3 and Wilkinson (n.d.(a)).

  7. Although generations are commonly used as locations in the social welfare literature, I agree with Askell (2018, pp. 12–13) that they are typically underspecified and, when precisified, bring implausible results.

  8. This set may be countable or uncountable. If the latter, we won’t be able to list our locations with natural-numbered subscripts. But much of what follows does not depend on which it is.

    If our locations are persons, we can assume that the set will be countable, for reasons similar to those raised in Footnote 2: persons require some positive, finite volume and duration. We could set some minimum physical volume and duration for a person to matter morally, and only a countable infinity of persons will fit into any spacetime (whether Euclidean or not).

    If our locations are spacetime positions, we can assume that the set will be uncountable. After all, these are positioned densely.

  9. I am assuming that spacetime is Euclidean: that points have a unique coordinate representation up to translation and scalar multiplication; and that every two points have a unique inner product and geometric distance between them (up to scalar multiplication). This distinguishes the spacetime I’m working with from Minkowskian and pseudo-Riemannian spacetimes which are more accurate representations of our universe and which each admit different coordinate representations under which inner products and geometric distance are not fixed. I accept that the assumption of Euclidean spacetime is unrealistic, but necessary to avoid taking us too far afield. I wish to demonstrate that my method overcomes basic problems in even the Euclidean setting before attempting to deal with problems introduced by relativistic spacetimes. In further work, I show that these further problems can be resolved as well (Wilkinson n.d.(b)).

  10. Note that I am not committing myself to a particular position in the debate—between Lewis (1971) and, for instance, Kripke (1980, p. 45)—on whether transworld identities exist. The following relation works just as well whether it’s an identity relation or just a privileged counterpart relation.

  11. This clearly holds in Euclidean spacetime. In relativistic spacetimes, the counterpart relation becomes more complicated (see Wilkinson n.d.(b)).

  12. Note also that this proposal does not require that counterpart positions are metaphysically identical, nor that they have essential properties. I propose merely that, in our moral evaluations, we treat points as equivalent when they satisfy these relations. This is compatible with both spacetime substantivalism and relationism.

  13. Transitivity has its critics, e.g., Temkin (2012). I nonetheless find it overwhelmingly plausible. In keeping with the infinite aggregation literature to date (within both moral philosophy and welfare economics), I will assume without argument that it holds.

  14. Lauwers shows that the following impossibility holds even for the domain of worlds with local values of only 0 and 1.

  15. It has been shown that such relations do exist, but cannot be constructed (see Svensson 1980). This is unsatisfactory for an moral rule, by which we would like to explicitly define the criteria for betterness.

  16. These results are originally given in a setting in which we apply Pareto and Finite Anonymity over generations. But as Askell (2018, pp. 39–44) explains, with some minimal assumptions, we obtain the same result if we take persons as locations. Likewise, it applies if spacetime positions are locations.

  17. Cases like this, which demonstrate the disagreements between different Pareto principles, originate with Cain (1995, pp. 401–403). Cain considers them a decisive objection to spatiotemporal views like mine. But, as we’ll see below, views that endorse Pareto over persons face problems that you might consider even worse.

  18. This assumption is often called Relative Anonymity in the social welfare literature, e.g., by Asheim et al. (2010, p. 10).

  19. The preceding argument is loosely adapted from Askell (ibid.), although Askell accepts this incomparability rather than abandon Pareto over persons.

  20. Despite the conflict between the two forms of Pareto, Pareto over persons does still have some intuitive draw. Given this, it may seem desirable to have \(W_a \succcurlyeq W_b\) (or \(W_a \succ W_b\)) whenever Pareto over persons says so and Pareto over positions does not deny it. But Firing Line demonstrates that this can bring problems. In Firing Line, Pareto over persons says that \(W_3\succ W_2\) and \(W_1\succ W_4\) and Pareto over positions doesn’t deny it, but that leads us to judging \(W_1\) and \(W_2\) as incomparable.

    Given this, we must be cautious of following the verdicts of Pareto over persons even in narrow circumstances. But, I will briefly note, one situation in which it seems harmless to do so is when, across the two worlds we are comparing, only finitely many persons have their spatiotemporal positions differ between worlds. And my own proposal below satisfies Pareto over persons in those cases.

    Another situation in which it seems harmless is when Pareto over positions (or some strengthening of it) says that two worlds are equally good. If so, and if Pareto over persons says that one world is strictly better, we might break the tie according to whatever Pareto over persons recommends. My own proposal doesn’t do this, but it does seem a plausible modification to make.

  21. We might instead use a stronger notion of containment: a set A is larger than B if \(A\backslash B\) has greater cardinality than \(B \backslash A\).

  22. This is an application of asymptotic density to this particular context (see Halberstam and Roth 1966).

  23. Another candidate is numerosity, as described by Benci and Di Nasso (2003), which is an extension of containment.

  24. Arntzenius (2014) accepts a similar expansionist principle, designed to deal with cases of uncertainty. In cases of certainty, his principle is equivalent to SBI3.

  25. See Wilkinson (n.d.(a)) for a full discussion of these points.

  26. Define addition of worlds as follows. The world \(W_a=W_b+W_c\) is the world defined by \(V_a(l)=V_b(l)+V_c(l)\) for all \(l\in {\mathcal {L}}\) and \({\mathcal {L}}_a={\mathcal {L}}_b ={\mathcal {L}}_c={\mathcal {L}}\).

  27. Geometric distance may seem the most natural way to do this, but it is not the only way. Since time and spatial distance measure fundamentally different quantities, and use different units, there are many different ways we might choose to combine the two into a common metric. But to allow us to construct useful expansions, such a metric does need to satisfy three conditions. (i) For every pair of points, the distance between them must be defined (or else expansions from one point will never reach the other point). (ii) If we start with a bounded region, any uniform expansion of that region must also be bounded (or else our cumulative sums may be infinite). And (iii) Distance must increase for increasing changes in space and increasing changes in time, so that the expansions become larger in the natural sense. Together, these conditions imply that our distance metric must satisfy \(d^p=|\frac{\Delta x}{a}|^p+|\frac{\Delta t}{b}|^p\) for some positive, real ab and n. (For instance, Arntzenius (2014: 44) describes an approach which effectively uses \(a=b=1\).) If we apply an additional condition—(iv) that regions of fixed distance are invariant under rotation—then we obtain geometric distance.

    Note that, in a non-Euclidean space, there may be no definition of distance which satisfies (i)–(iii), let alone (i)–(iv), so we would need to abandon one of them. Wilkinson (n.d.(b)) sacrifices requirement (iii) to obtain a different distance metric. But even with the distance metric described there, we would obtain the same verdicts in the problem cases below. Likewise, if we adopted any metric which satisfies \(d^p= |\frac{\Delta x}{a}|^p+|\frac{\Delta t}{b}|^p\), we would obtain the same verdicts, so we can simplify the discussion by using geometric distance.

  28. In practice, these sets will always be the same if we use positions as locations. If we use persons merely positioned in spacetime, they may not be.

  29. Note that SE1, and other expansionist approaches, can be expressed in terms of linearly ordered abelian groups and filters. Pivato (2014) shows this for all \(\succcurlyeq\) relations which satisfy basic conditions resembling Finite Anonymity and Pareto. I won’t attempt to express SE1 in such terms here, but note that it is not necessarily inconsistent with approaches which use such representations (e.g., Bostrom 2011, pp. 20–24).

  30. This holds even though the delay between reaching the − 1 point and the partner + 1 point approaches 0. To see this, simply compare the distance from P to a − 1 point to the distance from P to its partner + 1 point—the latter is always strictly greater.

  31. Could SE1 be modified to avoid this verdict, while leaving intact its core features? Yes, but such modifications bring on other nasty implications.

    One option is to abandon the requirement that expansions be uniform. For instance, we might replace them with sequences of square-shaped (hypercube-shaped) regions, as Askell (2018, pp. 208–11) does. This alternative may avoid the problem in Christmas, but it still faces the problems below. I should also note that this solution also appears somewhat arbitrary—it remains sensitive to some changes in \(\Delta x\) and \(\Delta t\) but not others.

    Here is another option, indeed the only option if we hold fixed the uniform expansions and the dependence of \(\succcurlyeq\) on the cumulative sums over those expansions. When making judgments, do not require that all starting points agree on judgments. Instead of supervaluating over starting points, we could subvaluate. After all, in the Christmas case, there is some starting point (R) for which the cumulative sums of \(W_0\) and \(W_1\) are equal for all expansions (and there are none which give a conflicting verdict). Why not then judge them as equally good? Upon examination, it turns out that doing so results in a \(\succcurlyeq\) relation which is intransitive.

  32. As above, this simplification is allowed by the assumption that value is additively separable.

  33. I am indebted to Christian Tarsney for raising this problem.

  34. To make matters worse, by helping the person cross the road and thereby altering the position of your body for a short while, you’ll ever-so-slightly change the gravitational tidal forces acting on other astronomical objects. Over timescales of a more than 500 million years, this significantly alters the positions of planets in our solar system, and thereby changes the details of many events that happen thereafter (see Wilkinson n.d.(a)).

  35. This relates closely to the problem of simple cluelessness for objective betterness (see Greaves 2016). In the standard problem, one of two actions produces an initial benefit of k units of value, but we are clueless about the future impacts of both actions. Over long time horizons, the probability of either action turning out better approaches 0.5. So we are clueless of which is better. But in infinite cases like Writing or Netflix, by SE1, that probability drops to 0—we are practically guaranteed that the resulting worlds are incomparable.

  36. SE2 can be stated more generally to allow for value spread densely over a region rather than concentrated at discrete points. To do so, we need only replace the sum with an integral: \(\int _{0}^{\infty} \sum _{{\mathbf{x}} \in E(r, P)} V_{1}({\mathbf{x}})-V_{2}({\mathbf{x}}) dr.\)

  37. Admittedly, there’s a probability of \(\frac{1}{2}\) that \(W_0\) is the better world, so Rita would still be very uncertain of which act would turn out better. But this is just the same conclusion we face in the finite case (Greaves 2016).

  38. I’ve described how we can restore axiological judgments. I haven’t proposed a way to restore our judgments of objective or subjective normativity. Nonetheless, extending this view to objective normativity is straightforward. And Wilkinson (n.d.(c)) provides a suitable extension to subjective normativity.


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This paper has benefited enormously from the input of many brilliant and generous colleagues. I am indebted to Christian Tarsney, Peter Vallentyne, Teru Thomas, and especially Alan Hájek for each giving their frank and thorough feedback on drafts. I am grateful to Andreas Mogensen, Marc Fleurbaey, Adam Jonsson, Ralf Bader, Timothy L. Williamson, and to a seminar audience at the Princeton Philosophy Society for valuable discussion. On the logistical side, I am grateful the Princeton Department of Philosophy for allowing me to visit for much of the 2018-19 academic year, and the Australian-American Fulbright Commission for funding that visit, during which time this paper took shape. This work was also funded by both the Australian Government Research Training Program and the Forethought Foundation for Global Priorities Research, via a Global Priorities Fellowship; my thanks go to both funders for their support.

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Wilkinson, H. Infinite aggregation: expanded addition. Philos Stud 178, 1917–1949 (2021).

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