Population ethics in an infinite universe

If you wish to make an apple pie from scratch, you must first invent the universe.

—Carl Sagan.

Abstract

Population ethics studies the tradeoff between the total number of people who will ever live, and their quality of life. But widely accepted theories in modern cosmology say that spacetime is probably infinite. In this case, its population is also probably infinite, so the quantity/quality tradeoff of population ethics is no longer meaningful. Instead, we face the problem of how to ethically evaluate an infinite population of people dispersed throughout time and space. I argue that axiologies based on spatiotemporal Cesàro average utility are the most appropriate way to make this evaluation, because their unique properties make them superior to other axiologies that have been proposed.

Appendices

Appendices

A Against exponentially discounted utility sums

In economics, the standard way to evaluate an infinite future stream of utility is by an exponentially discounted sum. Of course, discounting the interests of some people just because they have the misfortune to be born later than others can be challenged on purely moral grounds. In Ramsey’s (1928) famous words, it is “... a practice which is ethically indefensible and arises merely from the weakness of the imagination.”⁴⁹ Nevertheless, exponential discounting is ubiquitous in economics because of its ease and practicality. So it is worth explaining why this approach cannot work if we take modern physics seriously.⁵⁰

One problem is that exponential discounting is typically applied to moments in time, whereas we must also aggregate utilities across space. There are two natural responses: (i) confine attention only to the future light cone of the observer; and (ii) apply exponential discounting with respect to spatial as well as temporal distances. However, I shall now explain why neither solution is viable.

According to relativistic physics, no signal or other causal influence can travel faster than the speed of light. Consider an observer who is situated at some point o in spacetime. Her future light cone is the set of all points in spacetime that she can reach from o with a signal travelling at light speed or slower, as shown in Figure 7(a). This is the set of all locations in future spacetime where she can exert any causal influence —hence, arguably, the set of all locations relevant to a consequentialist moral evaluation of her present actions.

Fig. 7

A sketch of relativistic spacetime. Heuristically, the vertical axis represents “time” in these figures, while the horizontal axis represents “space”. But this is only a heuristic: in relativistic physics there is no clear distinction between space and time. (a) Future and past light cones through the point o. The lines represent the paths of possible signals through spacetime. (b) Curves of constant distance from o in the Minkowski metric. Note that Minkowski distances are only well-defined inside the light cones

So it seems that she should just apply exponential discounting to events in her future light cone. To do this, she must assign a “time” coordinate to each point in her future light cone. But according to relativistic physics, there is no objective time coordinate. As shown in Figures 4 and 5 in Sect. 3, two observers with different velocities will assign different time coordinates to the future, and thus exponentially discount future utilities in different ways. So they will disagree in their ethical evaluations —even if they use the same discount factor. (See also the bottom row of Figure 6 in Sect. 3.)

There is a further problem. The main justification for exponential discounting (since Koopmans, 1960) is Stationarity: if the utility stream $(u_0,v_1,v_2,v_3,\ldots )$ is preferred to stream $(u_0,w_1,w_2,w_3,\ldots )$, then $(v_1,v_2,v_3,\ldots )$ should be preferred to $(w_1,w_2,w_3,\ldots )$. This axiom of “intertemporal consistency” prevents an observer from gratuitously reversing her preferences as time passes. But in relativistic spacetime, Stationarity is impossible: whenever the observer changes her velocity, her reference frame changes with it, in a manner which violates Stationarity even if she is an exponential discounter. So an observer can satisfy Stationarity only if she travels in a straight line at a constant speed, forever.

For the small velocity differences between different observers on planet Earth, and over the small spatiotemporal scale of everyday decisions, these interpersonal disagreements and intrapersonal deviations from Stationarity are negligible. But when extrapolated over the entire infinite future, they can be quite significant —even for small velocity differences. And for observers with large velocity differences (who may exist in the future), these disagreements and deviations can be significant even on small spatiotemporal scales.

A partial solution is a different form of discounting, which I shall call Minkowski discounting. This means that each observer exponentially discounts each spacetime location s in her future lightcone according to Minkowski metric distance from her current location to s. The Minkowski metric measures how much subjective time (or proper time) she would experience if she travelled from her current position to s in a straight line at a constant speed; see Figure 7(b).⁵¹ The advantage of Minkowski discounting is that Minkowski distances are independent of the observer’s reference frame. So if Alice and Bob are both at the same spacetime position, they will Minkowski-discount events at s by the same amount, even if they have different velocities. So Alice and Bob will agree on their ethical evaluations. But the disadvantage of Minkowski discounting is that it violates Stationarity even for observers travelling at a constant speed. (This can be seen by shifting upwards the parabolas in Figure 7(b).) Likewise, two observers at different points in spacetime will Minkowski-discount the future differently, even if they have the same velocity.⁵²

In short, it should be clear that proposal (i) is unworkable. But now the problems with proposal (ii) should also be clear. In addition to being even more “ethically indefensible” than temporal discounting, spatial discounting faces the problem that in a relativistic spacetime, there is no observer-independent way to measure spatiotemporal distances. Different observers, travelling at different velocities, will have different reference frames, hence assign coordinates to spacetime in different ways, resulting in different measurements of distance.⁵³ Furthermore, the Minkowski metric is of no help. Although it assigns positive “distances” to points in the interior of an observer’s light cones, it assigns zero distance to any point on their boundaries (i.e. along the trajectory of photons) and assigns imaginary number distances to points outside the light cone. So using the Minkowski metric for spatial discounting would yield absurdities.

The upshot of this analysis is that exponential discounting is a nonstarter for any ethical theory which is intended to apply over large stretches of space and time.⁵⁴ This provides another reason (beyond the purely ethical argument) to seek a nondiscounted way of aggregating the utilities of people scattered across space and time.

B Against utility density

Recall the terminology of Sect. 2, where ${\mathcal { S}}$ is spacetime equipped with a metric d, and c is a “centre” point in ${\mathcal { S}}$. For any positive real number r, let ${\mathcal { S}}(c,r)$ be the ball of radius r around c in ${\mathcal { S}}$.⁵⁵ Let V(r) be the volume of this ball.⁵⁶ The set ${\mathcal { N}}_c({\mathbf{ s}},r)$ defined in Sect. 2 is just the set of people whose spatiotemporal locations are inside of ${\mathcal { S}}(c,r)$. As in Sect. 4.3, let ${\mathcal { X}}$ be a set of life outcomes. Let $({\mathbf{ s}},{\mathbf{ x}})$ be a world, and let u be a function from ${\mathcal { X}}$ into ${\mathbb {R}}$, assigning a utility to every possible life outcome in ${\mathcal { X}}$. The utility density of this world is the limit, as r becomes very large, of the total utility of all people contained in a ball of radius r, divided by its volume.⁵⁷ (Bostrom (2011, §2.3) calls this value density.) As already noted in Sect. 1, an axiology based on utility density yields a population density version of the Repugnant Conclusion. Also, the definition of utility density assumes that there is an objective, observer-independent way to measure distances in spacetime —an assumption already refuted in Sect. 3 and Appendix A. But this axiology is also untenable for several other reasons. First, consider an infinite universe with only a finite number of people. In such a universe, the utility density would always be zero, regardless of the quality of people’s lives. Thus, a utility density axiology effectively requires an infinite population. Although I have earlier argued that an infinite population is the most likely world, it would be strange if our axiology depended on such a hypothesis to be nontrivial.⁵⁸ Alternately, suppose the population is infinite, but confined to a two-dimensional subset of three-dimensional space. In such a universe, the utility density must be zero, regardless of the quality of people’s lives. This is absurd.

Furthermore, our best scientific theories say that the universe will expand forever, so the spatial density of matter and energy is will decrease to zero over time. On the plausible assumption that all forms of life require some substrate of matter and energy, this means the density of people must also decrease to zero over time. Since the balls considered in the definition of utility density expand in time as well as in space, they will eventually be mostly occupied by the mostly empty spacetime of the far future; this again implies that the utility density must be zero, regardless of people’s wellbeing. This is absurd.

Of course, our theories may be wrong. If the mass-energy density of the universe is high enough, then its expansion will eventually halt and go into reverse, which would obviate the problem in the previous paragraph. But it would be very odd if the viability of our axiology depended upon whether the cosmos exceeds a certain critical density.

Fig. 8

Left. A two-dimensional manifold equipped with a metric giving it the geometry of an open disk in a flat Euclidean plane, with finite area. Right. The same two-dimensional manifold equipped with a different metric, giving it the geometry of an unbounded hyperbolic plane, with infinite area. Every triangle in the picture has the same area. (Image credit: Anton Sherwood)

But this is just one symptom of a much deeper problem. The definition of utility density takes for granted that there is some objective way to define the “volume” of a region in spacetime. There is a natural notion of “volume” in familiar three-dimensional Euclidean space, and also in the four-dimensional Minkowski spacetime underlying special relativity (the Lebesgue measure). In both cases, it arises from the underlying geometric structure of the space (its inner product structure). However, in general relativity, spacetime is a four-dimensional differentiable manifold without any intrinsic geometric structure. The geometric structure of relativistic spacetime is entirely determined by its distribution of matter and energy.⁵⁹ Different mass-energy distributions yield different geometries, hence different measures of volume.

For a simple (non-relativistic) example, consider Figure 8. This shows a two-dimensional manifold (i.e. a surface) equipped with two different metrics. In the left-hand image, the manifold has a metric giving it the geometry of an open disk in a flat Euclidean plane. In the right-hand image, the same manifold has a different metric, giving it the geometry of a hyperbolic plane, which has infinite area. The relevant examples in general relativity are four-dimensional, and furthermore involve a Lorentzian metric. But the basic point remains the same: volume is determined by the geometry of the manifold, and in general relativity, this geometry is not fixed in advance.

This is problematic, because the volume V(r) plays a crucial role in the definition of utility density. There can be two universes, with exactly same people, each having exactly the same quality of life and occupying exactly the same “positions” on the underlying spacetime manifold (i.e. the same world $({\mathbf{ s}},{\mathbf{ x}})$), but with different distributions of matter and energy, leading to different geometries and hence different values for the utility density. Worse: a modification to $({\mathbf{ s}},{\mathbf{ x}})$ which increases utility density in one universe might decrease utility density in the other universe. Hence whether or not this modification counts as an improvement would depend on the spatiotemporal distribution of (non-living) matter and energy. This is obviously absurd.

C Index of notation

$\succcurlyeq$:

A weak order (i.e. complete, transitive binary relation) over worlds, representing an axiology. Its symmetric part is denoted $\approx$, and its antisymmetric part is denoted $\succ$.

$\succcurlyeq _*$:

A weak order over individual life outcomes, induced by $\succcurlyeq$. Its symmetric part is denoted $\approx _*$, and its antisymmetric part is denoted $\succ _*$.

c:

A point in spacetime ${\mathcal { S}}$ used as a “centre” for computing spatiotemporal Cesàro average utilities.

d:

A metric on ${\mathcal { X}}$, and also a metric on ${\mathcal { S}}$.

$\delta$:

A metric on the set $\Phi$ of reference frames; see §4.1.

$d_{\infty }$:

The supremum metric induced by d on the set of social outcomes; see Footnotes 11 and 39.

$\eta :{\mathcal { N}}{{\longrightarrow }}{\mathbb {N}}$:

An “enumeration” of people according to their locations in spacetime.

${\varvec{\mathcal {K}}}=\{{\mathcal { K}}_n\}_{n\in {\mathcal { N}}}$:

A soma, indicating the spatiotemporal location and extension of every person. Here, ${\mathcal { K}}_n\subseteq {\mathcal { O}}$ for all $n\in {\mathcal { N}}$; see §4.2.

${\varvec{\mathcal {L}}}=\{{\mathcal { L}}_n\}_{n\in {\mathcal { N}}}$:

Another soma.

${\mathcal { N}}$:

A countably infinite indexing set, used to label all the people who will ever live in the universe.

${\mathcal { N}}_{c,\psi }({\varvec{\mathcal {K}}},r)$:

The set of all indices of people whose spacetime location appears to be contained in a ball of radius r around the center point c, with respect to some reference frame in a ball of radius r around the central reference frame $\psi$; see Footnote 28 in §4.2.

${\mathbb {N}}$:

$:=\{1,2,3,\ldots \}$ The set of natural numbers.

$n,m,\ldots$:

Generic elements of ${\mathcal { N}}$, or generic natural numbers.

${\mathcal { O}}$:

A set representing objective spacetime; see §4.1.

$\Phi$:

The set of possible reference frames; see §4.1.

$\Phi (\psi ,r)$:

The set of all reference frames in a ball of radius r around the central reference frame $\psi$; see Footnote 28 in §4.2.

$\phi ,\phi ',\ldots$:

Generic reference frames; see §4.1.

$\psi$:

A reference frame in $\Phi$ used as a “central” reference frame for computing spatiotemporal Cesàro average utilities.

${\mathbb {R}}$:

The set of all real numbers.

${\mathbb {R}}_+$:

The set of all positive real numbers.

$r,R,\ldots$:

Generic real numbers

$r_{c,\psi }(n)$:

The smallest radius r such that person n appears to be contained in a ball of radius r around the center point c, with respect to some reference frame in a ball of radius r around the central reference frame $\psi$.

${\mathcal { S}}$:

A set representing subjectively perceived spacetime; see §4.1 and the start of §2.

${\mathcal { S}}(c,r)$:

The set of all points in subjective spacetime ${\mathcal { S}}$ within distance r of the centre point c; see Footnote 28 in §4.2.

$s,s'\ldots$:

Generic elements of ${\mathcal { S}}$.

${\mathbf{ s}},{\mathbf{ s}}',\ldots$:

Generic spatiotemporal arrangements —that is, sequencess $(s_1,s_2,s_3,\ldots )$ that assign a spacetime location $s_n$ in ${\mathcal { S}}$ to every index n in ${\mathcal { N}}$; see the start of §2.

${\mathcal { W}}$:

The set of all nomologically possible worlds; see the end of §4.4.

$u:{\mathcal { X}}{{\longrightarrow }}{\mathbb {R}}$:

A “utility” function. It assigns a numerical value to every possible life outcome, measuring the overall quality of that life for the purposes of ethical evaluation.

${\mathcal { X}}$:

A locally compact metric space, representing the set of all possible individual life outcomes; see §4.3 and the start of §2.

$x,y,z,\ldots$:

Generic life outcomes —i.e., elements of ${\mathcal { X}}$.

${\mathbf{ x}},{\mathbf{ y}},{\mathbf{ z}},\ldots$:

Generic social outcomes.