*by Michael C. Dorf*

*The Supreme Court Term is over, while the new one hasn't begun. There's plenty of legal news to blog about, most of it terrible, but it's summer, and I need a break. Hence, today I shall discuss a topic unrelated to law: How should I feel about an infinite multiverse? I realize that's a bizarre question, so I'll begin with the backstory of how I got interested in it.*

For all but one of the last 11 summers, Prof Colb and I have been attending what is now called Vegan Summerfest, a five-day conference on ethical, environmental, and nutritional issues that arise out of the exploitation of non-human animals for food and other purposes. When we first started attending we were there as listeners, but for the last several years we have been featured speakers. At the most recent gathering, Prof Colb gave two solo talks, I gave one solo talk, and we gave one together. I also attended a variety of interesting talks and panels.

As one would expect from an extended conference, much of the action occurs in informal settings, especially at mealtimes. Often these conversations end up continuing discussions begun in the sessions. Because I'm drawn especially to the ethics sessions, I tend to try to continue the discussions involving ethics. One question arose for me out of a session in which a speaker was using a utilitarian framework: If the universe is infinite and contains an infinite number of beings with on-average positive utility, how does a total-utility utilitarian decide which acts are better than others?

Let's suppose I'm deciding whether to spend $100 on some frivolous pleasure or to donate it to a non-profit that does a good job of fighting poverty. My frivolous pleasure generates for me one utile but the non-profit will use it to generate 100 utiles. In a finite world, the donation is the clearly better choice, but not in an infinite world, because infinity plus 100 is still infinity. So if we live in an infinite universe with infinitely many utility-bearing beings of average positive utility, utilitarianism would appear to break down. (One gets the same result if one assumes average negative utility, because negative-infinity plus or minus any finite number is negative-infinity, rendering one's efforts at amelioration futile.)

To my surprise, I could not get anyone interested in discussing the difficulty utilitarianism has with infinity. Some of my interlocutors even said this was not a real problem--which is obviously false, because I later looked it up and discovered that there are many philosophy papers discussing the problem infinity poses for utilitarianism. (E.g., this one, this one, and this one). My discovery of this literature (which, to be clear, I discovered with no effort beyond Googling) was both encouraging and discouraging. I was encouraged that I had stumbled upon what philosophers consider a genuine rather than a silly problem. I was discouraged that I wasn't the first person (in our corner of the universe) to have noticed the problem.

I could not interest my fellow conference-goers in the problem infinity poses for utilitarianism, but I might have if the subject hadn't changed. One attendee with whom I've become friends is a physicist, who, a propos of my invocation of infinity, opined that he thinks the

*many-worlds*hypothesis is probably true. This caused me a familiar unsettling feeling.

I probably don't need to do this for anyone who's still reading, but just in case you don't know what the many-worlds hypothesis is, here's a vast oversimplification of the idea: Our best depiction of the subatomic world does not have particles at discrete locations but rather has wavelike probability distributions for finding particles at any given point. When an observer measures a particle's location, the wave function collapses to a point. But why does it collapse where it does? The many worlds hypothesis says that there is no collapse, that every possibility is realized, but each discrete realization creates a different "branch" of the world. So gazillions of particles are constantly branching, creating gazillions of gazillions of different possible worlds.

*Many worlds*says that they're all real, but we just inhabit our one branch.

Okay, now that you know what the many worlds theory is, you might ask why I find it unsettling. A better question would be why I find the many-worlds infinity more unsettling than two other kinds of infinity. First, we might live in a spatially infinite universe, which means that if you go far enough in any direction, you'll eventually come to a region of space that contains a duplicate of you. And because infinity is REALLY BIG, you'll then come to another duplicate, and another, and so on. Mind-blowing? Absolutely. But somehow it's less disconcerting than the many-worlds universe.

A second kind of infinity has our universe as but one of an infinite number of universes in a multiverse, each with its own laws of physics and its own space-time. This too is mind-blowing but also somehow less disconcerting than many-worlds.

One reason why one might be more disconcerted by one kind of infinity than another is that some infinities are more infinite than others. Readers with sufficiently advanced math backgrounds will understand that statement as a reference to the work of the great 19th century mathematician Georg Cantor, who denominated different levels of infinity. To give the best known example, the number of even numbers is the same level of infinity as the number of whole numbers (even though it seems like there are twice as many whole numbers as even numbers, because the set of whole numbers includes odd as well as even numbers), but there are more real numbers than there are whole (or rational) numbers. If you didn't follow that, don't worry. The point is that some infinities are larger than others.

But I'm pretty sure that the reason I am more disconcerted by the many-worlds infinity than by a spatially infinite universe or a multiverse consisting of infinitely many so-called bubble universes is not that the many-worlds version corresponds to a higher order of infinity. I'm not actually sure it does.

So why do I find many-worlds more disconcerting? Partly it may have to do simply with the strangeness of the idea. If the universe extends infinitely in all directions, that doesn't much affect how I think about reality, even if it implies that there are (an infinite number of) duplicate me's out there far enough away. As the mother of the young Alvy Singer says in

*Annie Hall*in response to Alvy's having stopped doing his homework upon learning that the universe is expanding, "Brooklyn is not expanding." (That was before gentrification!)

By contrast, the idea that at every infinitesimal moment gazillions of alternative universes are coming into existence right next to us (as it were) seems much more immediate. What's worse, it feels like an occasion for regret. Sure, many of those alternative universes are terrible. There's one (or more likely gazillions) of universes in which the dinosaurs never went extinct and the Earth is inhabited by super-intelligent lizards. There's another (or gazillions) in which the Nazis won the war. But there are also gazillions of delightful parallel universes in which Donald Trump isn't president. Alas, we're stuck in this one.

## 5 comments:

From what I can think of, infinite time provides a much larger problem than infinite universe size, though they are related.

(For all of the below, I assume that faster-than-light information transfer is impossible. Without this assumption, it is possible for a single choice to affect the entire infinite universe at once, which breaks these ideas.)

If I assume finite time then I think there's a relatively simple solution to the infinite universe utility maximization problem.

For any choice, the highest utility action in the universe is the one that:

Define a finite set of the universe such that:

1.) All utility effects of the choice are included in that finite portion of the infinite universe.

2.) All areas of the infinite universe outside that finite portion are unaffected by the choice.

Given that set, the highest utility choice is the one that maximizes utility within that finite portion of the universe, since the remainder of the universe is unaffected.

This has the added bonus of being basically what we mean when we say maximizing utility, since at some point the scope of a choice is beyond our ability to measure, and thus we usually discount those effects.

Getting back to infinite time:

Infinite time creates a problem because with infinite time, it's possible that some choice has an infinite effect on utility. This brings in a larger problem, but I think we can use a similar trick. I'll also assume that utility has a time component, and thus can be integrated over time. There may be other ways to measure a time-based utility, but I will use integration.

Given that utility over time is integrable, then I would define the highest utility choice in an infinite universe as:

The highest utility choice in an infinite universe with infinite time is the choice such that for some time t, the integral of all utility effects of that choice over time between when the choice is made and all finite times greater than or equal to t, the net utility effect over the previously defined finite portion of the universe affected is greatest.

Now, it's possible that some choices will have a sinusoidal effect on utility over time, such that there is no such time t after which one choice is always better. In this case it may not be possible to define a "best" choice in a utilitarian way in even a finite universe that has infinite time. This is why I consider infinite time to be so much bigger of a problem than infinite space.

Here's my Ode that I might have earlier posted in a comment at this Blog:

***

MULTI-VERSES ON MULTIVERSES

There is no edge to our universe

As it expands, continues in infinity.

But what if there is a curse

Of multiverses with interstellar diversity?

Each with its own “Big Bang” birth

Of differing magnitudes,

With planets like our own earth

Peopled with conflicting attitudes.

Might they all function universally

Or each taking its own course

Competitively, independently

Each with its own forms of force?

Universe versus universe,

With their own clashing gravities,

Or forces even worse,

Like Spaceballs* banalities.

April 3, 2015

*For variation, substitute “Star Drek”.

***

When it comes to infinity, I am pi-eyed. And Greg, at 88 I don't have infinite time on my hands.

To me, the worst thing about many worlds is it eliminates free will. Every possible outcome occurs, so while I may have chosen alternative A, in another of the many worlds I have chosen B. Every possible choice has to be made, so the universe creates as many of 'me' as necessary to fulfill all paths.

On the bright side, everyone is immortal in the many worlds. People don't just die, they die for a reason. So as long as there is an alternative where it is possible you don't die, there is a 'you' to fulfill that branch.

With the expanding universe and multiverses our planet Earth will become more and more "infinit[y]esimal."

This reminds me of the four rabbis in the Talmud who discuss unanswerable questions with only one emerging sane from the discussion. Some things may be better avoided.

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