Summary

Is infinity real or just a fiction of our minds? I briefly review a few arguments for and against the existence of infinity from physics and philosophy, and I conclude that neither side is obviously right. A further question is whether we should prudentially believe in infinity because if it does exist, everything we do matters vastly more. Within a given ontological framework this argument is sound, but different ontological frameworks (e.g., ultrafinitism vs. transfinitism) are not directly comparable. I personally hope infinity does not exist so that the universe contains only finitely much suffering, but unfortunately I think it's reasonably likely that the universe is literally infinite. There remain numerous puzzles in how we approach the ontology and ethics of infinity, many of which may be best left to the future to sort out.

Introduction

Reducing suffering requires knowing what suffering exists, where, and in what quantities. Ontological investigation of physics and mathematics bears on what answers we give to those questions. In particular, it seems very relevant to know whether there's such a thing as infinity. Can there be infinitely many organisms? Can the universe exist for an infinite duration? How do we make ethical valuations when infinities are in play? This piece reviews a few broad ideas about the nature of infinity, but it's only an introduction to a much bigger subject.

Is there an ontological case for infinity?

Infinite physical quantities

When I was very young (around age 6), I was puzzled by the apparent infinity of the universe. I found the idea of infinity incomprehensible, and yet I reasoned: If the universe is finite, it has some boundary, but then what's outside that boundary? If you put the universe in a box, there's always space outside the box? Since then I've learned that this needn't be the case. For instance, the universe needn't be infinitely old because space-time may fold at the big bang, like the end of an egg, without anything existing outside of the timeless, bounded object that is the universe. So a finite universe isn't logically impossible.

Nonetheless, it appears empirically that "the shape of the universe is infinite and flat", with an unbounded future ahead of it. In addition, many multiverse theories predict infinitely many universes.

David Pearce points out that when infinities crop up in physics, they often cause trouble. For instance, cosmological singularities seem in conflict with quantum uncertainty. Renormalization and regularization attempt to patch infinite terms in quantum field theory and related subjects. (That said, even if human-constructed mathematics were to fail at describing physics, this wouldn't imply that physics wasn't infinite. Maybe it's just our problem for not being able to model what actually exists; the universe can ignore our clumsiness and keep on being whatever it is.)

It appears that infinite computation is not possible within a finite region of spacetime, which Scott Aaronson says may teach us "something not only about computation but also about physics." Dyson's eternal intelligence proposal for infinite computation looks unworkable because of a positive cosmological constant and a non-zero minimal vacuum temperature.

Indispensability argument

The classic defense of infinity's reality is the indispensability argument: Infinity is part of many scientific theories, so we should assume it exists. Penelope Maddy forcefully argues against this claim in Naturalism in Mathematics (p. 143):

If we open any physics text [...], the first thing we notice is that many of the applications of mathematics occur in the company of assumptions that we know to be literally false. For example, [...] we assume the ocean to be infinitely deep when we analyze the waves on its surface; we use continuous functions to represent quantities like energy, charge, and angular momentum, which know to be quantized; we take liquids to be continuous substances in fluid dynamics, despite atomic theory. On the face of it, an indispensability argument based on such an application of mathematics in science would be laughable: should we believe in the infinite because it plays an indispensable role in our best scientific account of water waves? [quote source]

In Mathematics and Reality, mathematical fictionalist Mary Leng discusses similar idealizations about infinity that are false if taken literally, such as "that fluids are continuous substances" (p. 111).

In 2007, I told one of my friends I was taking a second course in real analysis. He laughed and asked: "What is real analysis good for? Physics is discrete." I replied: "Continuous functions are useful for approximating discrete functions." It's kind of weird but true that math using infinities is often easier than finite math. But while it's easier to do calculus using infinitely small line segments, in principle calculus could stop at some large, finite level of granularity. My guess is that the same is true for most uses of infinity in the sciences. And even where infinity is indispensable, this doesn't mean it must exist ontologically; infinity may just be a helpful calculation trick for producing the right answers.

More fundamentally, infinity is never strictly necessary to explain our observations. Ben Goertzel points out:

all scientific data ever gathered, constitutes one large but finite set of bits (i.e a finite set of finite-precision numbers). Any finite set of bits can be modeled computationally. Of course, someone can claim a non-computable model is “better” than any computational model, for a given finite set of bits. But this then becomes a subjective claim, based on aesthetics, or intuition.

Scott Aaronson has discussed the finitude of our minds.

Max Tegmark believes that infinity probably doesn't exist. He notes:

Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. Our best computer simulations, accurately describing everything from the formation of galaxies to tomorrow’s weather to the masses of elementary particles, use only finite computer resources by treating everything as finite.

The main defense of infinity would be that it may produce more simple, elegant explanations of our observations. But it's debatable whether introducing a weird, arguably nonsensical idea like infinity into our ontology demands more or less of an Occam complexity penalty than developing more verbose finite formulas and algorithms. (Note that Solomonoff induction, a main tool of computer scientists for assigning Occam penalties, assumes the universe is the output of a Turing machine and so can't handle uncountable infinities.)

Suppose our multiverse is run on a computer in some basement universe. Eternal inflation predicts the emergence of infinitely many bubble universes, perhaps with different initial conditions. Naively, the simulating computer would run each new bubble universe separately. But suppose two bubble universes have the same initial conditions. Rather than simulating two identical copies, the computer might simulate just one and create two pointers to that single instance. Moreover, the computer might also simulate universes in a lazy fashion, allowing for potentially infinite objects to be constructed with finite resources.

Philosophies of mathematics

Platonists believe that mathematical objects actually exist in some ontological sense. So, if the set of real numbers, say, is a mathematical object, it must exist, and hence an actual infinity does exist.

I think mathematical platonism is confused and so don't buy this argument at all. Like Mary Leng, I subscribe to mathematical fictionalism. Playing around with symbols that represent infinities in consistent ways doesn't magically make infinities exist. You can also invent logically consistent stories in the Harry Potter universe.

Other philosophies of mathematics explicitly express skepticism about infinities. Intuitionism and finitism reject actual infinities, while ultrafinitism rejects even potential infinities.

The article "Philosophy of Mathematics" discusses one interesting weak argument for skepticism about real analysis. Halbach & Horsten 2005 note that from non-categoricity due to the Löwenheim-Skolem theorem Peano Arithmetic can be rescued by Tennenbaum's theorem if we insist that addition and multiplication be computable, which is a constraint consistent with what we know of physics anyway. But real analysis lacks an equivalent of Tennenbaum's theorem to save it from Löwenheim-Skolem. If we don't believe in real numbers in the first place, I assume we don't have this problem?

Modal realism

If one finds modal realism, ontological maximalism, or the mathematical universe hypothesis (MUH) compelling, do these imply infinitely many universes, including some universes that are infinite in size? This would seem to depend on how one defines a "possible world" or a "mathematical structure". For instance, an ultrafinitist MUH would be finite. Jürgen Schmidhuber has proposed a more constructive version of the MUH based on Turing-machine computation.

Is there a prudential case for infinity?

Suppose we're not certain whether physics is actually infinite, but we maintain an ontology that allows for the legitimate possibility of infinite physics. In this case, we should "act as if" the universe is infinite and we exist infinitely many times in it, because if so, our actions may have infinitely greater impact than if the universe is finite. This prudential reasoning seems legitimate, though it doesn't necessarily change which specific actions are recommended, because if infinity is real, any action has some chance of making an infinite difference. Indeed, if all possible configurations of matter exist infinitely many times, then exact copies of us exist infinitely many times, so anything we do has an infinite impact.

I think the prudential argument fails when applied across ontologies. For instance, suppose an ordinary mathematician, Max, is debating an ultrafinitist, Ursula:

Max: If you're right, you can only prevent finitely much suffering by your efforts. If I'm right, you might prevent infinite suffering. Thus, you should focus your efforts on the branch of possibilities where I'm right, because if I'm right, your actions would matter infinitely more. Ursula: What is this "infinity" whereof you speak? To me you're just talking gibberish. It would be as if I told you that it's possible squibberslop exists, and if it did, you could have squibbersloper more impact, which is way more important than just having infinite impact. So you should focus on scenarios where squibberslopism is true.

For any given ontology, there exists another ontology that can claim to be lexically more important.

This suggests that cross-ontology claims of superiority can't be rigorously defended; there's not enough common ground for one to make sense of the other. Nonetheless, I think it is important to challenge our ontological assumptions in non-rigorous ways, by the kinds of fuzzy reasoning and doubting of assumptions that human brains are good at. Exactly how to resolve fundamental epistemological and ontological disagreements is a tough question.

Infinity does not imply futility

In "Infinite Ethics", Nick Bostrom asks what the implications would be for consequentialism if the universe were infinite. An infinite universe would likely contain an infinite amount of suffering. In that case, would our reducing some finite amount of suffering make a difference? Infinity minus some finite quantity is still the same infinity.

Bostrom surveys many ingenious proposals for handling this difficulty. I'm personally more dedicated to the principle that reducing finite suffering is good regardless of the size of the universe than I am to any particular mathematical formalization of infinite ethics, so I'm not much bothered by the problem. But deciding how we feel about infinities may also have implications for less obvious moral dilemmas, so the issue does seem important. That said, it looks to me like physics solves Bostrom's particular puzzle, as discussed below.

Current physics deflates the futility argument

If physics is discrete, there are only finitely many possible configurations of physics of a given finite size (say, the size of our observable universe) and bounded temperature. Given some plausible assumptions, this means there must exist infinitely many exact copies of us. (Max Tegmark and Alexander Vilenkin have popularized this idea.) Moreover, a choice made by one of these exact copies logically implies what all the other copies choose, so if one copy, say, achieves victory in a humane-slaughter campaign here on Earth, infinitely many identical copies are achieving the exact same victory on parallel Earths. In other words, everything we choose makes an infinite difference.

This is also true if physics is continuous. What constitutes "you" is your decision-making algorithm, and that tends to be robustly instantiated despite changes in underlying details. (For example, certain parts of your body swap out their atoms over relatively short periods.) Since brain algorithms are apparently high-level, robust structures, what matters is just that the underlying physics is sufficiently similar to produce those algorithms. A "copy" of you is just another chunk of physics that exhibits the same behavior. It probably doesn't matter that much if an electron is 10-30 meters displaced in one copy relative to the other. (I'm pretending that electrons have unique locations just for illustration.)a

But how does the difference that our copies make compare with the infinity of all suffering in the universe? Because there are only finitely many relevantly different configurations of galaxy-sized chunks of physics, then assuming physics cycles through all configurations with a finite recurrence time (such as because of quantum randomness), then the fraction of all universe chunks that contain a copy of your galaxy is greater than zero. Since you can reduce a positive fraction of all suffering in your galaxy, this means the fraction of all suffering in the universe that your copies reduce by their actions is greater than zero. (I don't know if computing the exact fraction would depend on how limits are taken, but regardless, I would guess that under a sensible procedure for taking limits, your copies' fraction of all impact in the universe would remain greater than zero.)

Another way to see the same point is to remember that you have an exact copy roughly 101028 meters away. Thus, if we imagine computing the limit of the suffering your copies reduce as a fraction of total suffering in a sphere expanding from some point in space, the sphere would continue to contain copies of you roughly in direct proportion to its volume, so your fraction of the total would remain positive.

Moreover, this analysis hasn't even considered the partial correlations between similar but not identical minds. These imply even stronger benefits from altruism.

Bostrom himself recognized the above point in the "Class action" section of "Infinite Ethics". He notes that we still need further mathematical machinery to formalize consequentialist calculations. But at least we've established that infinity does not imply futility.

This example also explains why, as the prudential argument in the previous section asserted, the impact we can make on the universe is infinitely greater if the universe is infinite than if it's finite: It's because in that case there are infinitely many identical copies of us all doing good deeds.

I think the picture I painted is plausible given current mainstream physics. In particular, it seems consistent with any of Tegmark's Levels I, II, or III of multiverses, since these are actually all the same size. A Level I multiverse is arguably the standard picture in contemporary cosmology. My argument may or may not apply in a Level IV multiverse depending on what counts as a mathematical structure and what the measure is over different structures. In an ultrafinitist Level IV multiverse, there would be only finitely many possible universes, in which case infinity would still not be a problem. In a Level IV multiverse based on regular mathematics, there might be mathematical structures corresponding to all kinds of weird stuff, which could be uncountably infinite in variety, and if the weird stuff had big enough measure, it could render negligible by comparison the universe regions containing copies of us or even algorithms similar to ours.

Another case in which the measure of copies of us in the multiverse might be zero would be if the universe continues infinitely long into the future and doesn't produce Boltzmann brains. That said, unless we think quiescent de Sitter space is sentient, our copies would still comprise a nonzero fraction of all sentient physics in the universe.

Sizes and types of infinity

Cantor's cardinalities

At the end of my 12th-grade year in high school, my calculus teacher assigned the class to research topics in advanced mathematics at a non-technical level. My task was to report on Georg Cantor's transfinite numbers. When an English teacher asked what I was working on at the time, I said: "I'm learning about different sizes of infinity." The teacher laughed and replied something like: "I always thought mathematicians were extremely logical and reality-oriented, but I guess I was wrong."

Cantor's transfinite cardinals are probably the most widely known version of varying sizes of infinity. Most undergraduates in quantitative disciplines learn about the distinction between countable and uncountable sets. Cantor denoted levels of infinity by aleph numbers. For instance, "ℵ 0 " refers to the size of the set {1, 2, 3, ...}. With this number system, we can informally say that

ℵ 0 + 1 = ℵ 0 because, for example, {1, 2, 3, ...} has the same cardinality as {0, 1, 2, 3, ...}

+ 1 = ℵ because, for example, {1, 2, 3, ...} has the same cardinality as {0, 1, 2, 3, ...} ℵ 0 * 2 = ℵ 0 because, for example, {1, 2, 3, ...} has the same cardinality as {1, -1, 2, -2, 3, -3, ...}.

However, these facts do not mean that cardinal numbers are all the same size; for instance, the set of real numbers is strictly bigger than the set of counting numbers. Moreover, Cantor showed that the power set of any set is strictly bigger than the set itself. Taking power-set operations generates the beth numbers.

Some set theorists discuss large cardinals, though not everyone agrees with their legitimacy.

Infinity and religion

Eliezer Yudkowsky calls himself "an infinite set atheist". The "atheist" word captures an important idea: Belief in mathematical infinity feels a lot like belief in God, not just because God is often said to be infinite but also because infinity is a fantastical claim that can be talked about but doesn't necessarily correspond to anything actual. In general, platonist mathematicians who take their constructions seriously remind me of theologians building edifices of imagination.

Cantor himself was very religious and viewed his work as theologically relevant. He wrote in 1896 to a Dominican priest, "From me, Christian philosophy will be offered for the first time the true theory of the infinite" (qtd. in Bruce A. Hedmen, "Cantor's Concept of Infinity"). Cantor did not associate God with any particular transfinite number, but with the entire collection of them, which he denoted by the letter tav. According to the Wikipedia entry:

As Cantor realized, this collection could not itself have a cardinality, as this would lead to a paradox of the Burali-Forti type. Cantor instead said that it was an "inconsistent" collection which was absolutely infinite.

The Wikipedia article on absolute infinity explains:

The Absolute Infinite is mathematician Georg Cantor's concept of an "infinity" that transcended the transfinite numbers. Cantor equated the Absolute Infinite with God. He held that the Absolute Infinite had various mathematical properties, including that every property of the Absolute Infinite is also held by some smaller object.

Hedmen's article discusses this idea further:

Cantor [distinguished] between transfinite numbers, which exist in the human mind, and Absolute Infinity, which is beyond all human determination, and exists only in the mind of God. [...] Cantor thought of the infinite ascent of ever-increasing transfinite numbers as an appropriate symbol for the absolute.

Indeed, in an 1886 article, Constantin Gutberlet endeavored to defend Cantor's transfinite numbers on religious grounds:

But in the absolute mind the entire sequence [of transfinite numbers] is always in actual consciousness, without any possibility of increase in the knowledge or contemplation of a new member of the sequence. (qtd. in Joseph W. Dauben, "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite," p.100)

Interestingly, Christian apologist William Lane Craig is a prominent skeptic about actual infinity because this helps him make the Kalām cosmological argument.

Ordinals, hyperreals, surreals

While ℵ 0 doesn't change when finite numbers are added or multiplied to it, this is not the case for other constructions of infinity. Ordinal numbers, also developed by Cantor, "draw a finer distinction than cardinals on account of their order information", so that, for example, ω + 1 > ω and ω * 2 > ω. Ordinals can become massively big, such as with the limit ordinal ε 0 = ωωω..., the first uncountable ordinal ω 1 , then ω 2 , then eventually ω ω and ω ω ω and so on.

Other systems where ∞ + 1 > ∞ include John Horton Conway's surreal numbers and Abraham Robinson's hyperreal numbers. In the hyperreal number system, the standard infinimal, I, is identified with the sequence (1, 2, 3, 4, ...). Bostrom discusses arithmetic with hyperreals in "Infinite Ethics".

Which approach is right?

We can see that there are many ways to construct consistent mathematical infinities. Is one more "real" than others, assuming any infinity is real? I guess that depends on what's required for physical theories, if any infinities are required at all. Which should we use for ethical reasoning when making expected-value calculations? I think we should choose the infinity that best encapsulates our ethical intuitions. Indeed, if we have a strong intuition that's violated by the kind of mathematical infinity that we're using, maybe there's another infinity that accords better with our intuitions. For instance, hyperreal numbers accord better than transfinite cardinals with the idea that in an infinite universe, causing harm to one more organism increases total badness. (Thanks to a friend for inspiring this point.)

Infinite arms race?

As we've seen, many ways of constructing infinity imply that there's no biggest infinite number. Cantor's paradox shows this in the case of cardinal numbers. This means we can run into the problem of "my infinity is bigger than yours". Suppose we think action A would prevent ℵ 0 suffering with probability 1. However, there's some tiny but nonzero probability that action B would prevent ℵ 1 of suffering, so doing B seems to dominate in expectation. But maybe there's an even smaller chance that action A would prevent ℵ 2 of suffering, so we should do that after all. And so on.

For any given infinite cardinality, it seems that any action has some nonzero probability of reducing that cardinality of suffering. So maybe we should call off the arms race for bigger and bigger infinities and compare the actions on equal footing, i.e., relative to their impacts on a single, fixed cardinality of infinity?

Let future generations work out the details

Ontological questions surrounding infinity are perplexing. The sand can shift so quickly under our views on these topics that it doesn't make sense to take firm commitments to particular theories or implications at this point. It seems more effective to prepare the ground for future generations to better hash out these questions and to explore additional puzzles that we haven't even anticipated. This helps avoid Bostrom's "fanaticism problem" (p. 31) because many instrumental/strategic actions that target suffering reduction when only considering finite scenarios will translate to optimal actions when considering infinite scenarios.

Steering artificial general intelligence (AGI) in better directions seems like a more actionable short-term project than detailed exploration into the intricacies of infinite set theory, since if AGI goes right, it could further explore questions of infinity. That said, I think it is very important to be aware of how much remains to be sorted out in the realm of the ontology and ethics of infinity, so that we maintain adequate humility and avoid jumping to premature conclusions.