Virtue and Vice Among the Infinite

Here, I answer when infinities are possible, and when they’re impossible.*

[ *Based on: “Virtue and Vice Among the Infinite,” pp. 87-104 in Ad Infinitum: New Essays on Epistemological Infinitism, ed. John Turri and Peter Klein (Oxford University Press, 2014). Later elaborated in Approaching Infinity. ]

1. The Problem of the Impossible Infinite

There are many mind-boggling paradoxes involving infinity, but I can’t tell you all of them here. (There are 17 in my book.) Here, I’ll just tell you two that can be explained quickly:

a. Zeno’s dichotomy paradox

An object wants to move from point A to point B. Before it reaches point B, it must first go half the distance. Then, it must go half the remaining distance. Then half the remaining distance again. And so on. This is an infinite series. Therefore, the object can never reach point B.

That’s roughly how people usually phrase the problem. But how did we get the last step? Here’s a little more detail: Let “1/2” refer to going to the halfway point, “3/4” to going to the three-quarter mark, etc.:

1. To reach point B, the object must complete the series, [1/2, 3/4, 7/8, …].

2. [1/2, 3/4, 7/8, …] is an infinite series.

3. It is impossible to complete an infinite series. For:

   1. To complete a series, one must reach its end.

   2. An infinite series has no end.

   3. It is not possible to reach the end of a series that has no end.

4. Therefore, the object cannot complete the series [1/2, 3/4, 7/8, …]. (From 2, 3)

5. Therefore, the object cannot reach point B. (From 1, 4) But point B can be any arbitrary point (separated from A); therefore:

6. No object can go anywhere. (From 5)

The conclusion is crazy, since we know objects move. What went wrong?

b. Thompson’s Lamp

This example comes from James Thompson (Judith Thompson’s husband, in case you were wondering). There’s a lamp that starts out on at 12:00 noon. After 1/2 hour, someone switches it off. After another 1/4 hour, it gets switched back on. Then off after 1/8 hour, on after 1/16 hour, etc. Q: At 1:00 p.m., will the lamp be on or off?

It seems that there should be an answer to this, but no answer seems to be correct. Thompson offers two arguments:

• The lamp started out on, and it was never switched off without subsequently being switched back on again. Therefore, it must be on at the end.

• The lamp was switched off at 12:30, and thereafter, it was never switched on without subsequently being switched back off. Therefore, it must be off at the end.

So there are seemingly compelling argument for opposite answers. Thompson thinks this shows that the lamp scenario is (metaphysically) impossible.

Alternate answer: You could say it’s indeterminate. The description of the scenario only discusses what happened to the lamp between 12 and 1, not what happened exactly at 1:00. So, logically, the lamp could do anything, be in any state, at 1:00 p.m., consistent with everything in the description. So it’s indeterminate. (Benacerraf’s take, which Thompson later wrongly accepted.)

Reply: No one would find such a claim satisfactory for any normal counterfactual question. Normally, when asked “what would happen” in some scenario, we do not think that the description of the scenario has to entail the answer or else it’s indeterminate.

Ex.: Suppose there is a normal lamp that starts out on. Someone comes in and shuts it off. No one and nothing else does anything else to it. One hour later, what state would this lamp be in?

Notice that the description does not logically entail that the lamp would be off. It’s logically possible that the lamp be on, or off, or that it explodes, or that it turns into a bat and flies away. But we know what would happen. What would happen is that it would be off.

If you give a complete description of the lamp’s initial state and everything that is done to it in some time period, that normally enables you to say what state it would be in at the end. It doesn’t matter that it is logically possible for it to be in any state at the end.

c. The meta-puzzle

So you just read about two paradoxes. Here’s another puzzle that results from comparing those paradoxes:

• The solution to Zeno’s Paradox seems to be to reject (3). I.e., to argue that it is possible to complete an infinite series, since we do this every time we move.

• The solution to Thompson’s Lamp seems to be that the lamp is impossible, as Thompson suggested. Why? It appears to be because it is impossible to complete an infinite series.

I.e. the answer to the first paradox is the negation of the answer to the second paradox.

Well, presumably not. Rather, this pair of examples shows that some infinite series are completable, while other infinite series are not. Why is the Zeno series completable but the Thompson Lamp series not? The two series are very similar to each other on their face, so this is a puzzle.

2. The Impossible Infinite

a. Other infinities

There are other kinds of infinity, besides infinite series; there are infinite magnitudes. Some of these are widely regarded as problematic, while others are not. For instance, the center of a black hole is, per general relativity, a singularity, a point of infinite energy density, and hence infinite spacetime curvature. This is widely regarded as a problem, and part of what research into quantum gravity aims to do is to eliminate these infinities.

On the other hand, no one seems to have a problem with the idea of space being infinitely extended, or infinitely divisible, or time being infinitely extended and infinitely divisible. Nor does anyone have a problem with the idea of there being infinitely many numbers, or an infinite collection of parallel universes, etc.

Again, there’s a puzzle: Why are some infinities problematic, while others are not?

b. Natural intensive magnitudes

These puzzles have one basic answer: It is impossible for anything to have an infinite natural, intensive magnitude.

There are three kinds of quantities (things that are represented by numbers):

• Cardinal numbers: These are counting numbers; they are answers to a “how many” question. E.g., 1, 2, 3, and so on.

• Magnitudes: These are the kind of quantities represented by real numbers; they are answers to a “how much” question. These divide into 2 sub-kinds:

  • Extensive Magnitudes: These are basically additive magnitudes. E.g., length, duration, mass. If the left half of a stick is 1 foot, and the right half is also 1 foot, then the whole stick is 2 feet.

  • Intensive magnitudes: These are non-additive. E.g., temperature, mass density. If the left half of your cup of coffee is 120 degrees, and the right half is 120 degrees, it is not the case that the cup is 240 degrees.

A set can have an infinite cardinality, and a thing can have an infinite extensive magnitude. But nothing can have an infinite intensive magnitude. This explains the above cases:

1. Thompson’s lamp: Every time you switch the lamp, you need to move the switch faster than the last time. As a result, the energy required increases without bound. To complete the whole infinite series, you need a spacetime region with infinite energy density. In addition, the switch must be made of a material with infinite material strength. Both of these are intensive magnitudes, so this is impossible.

2. Zeno’s paradox: Zeno’s series contains an infinite number of steps, but this is only an infinite cardinality. All the physical magnitudes in the story remain finite. The energy, velocity, distance, duration, etc., are all finite. There’s no infinite intensive magnitude, so there’s no problem.

3. Singularities: These have infinite energy density, which is impossible.

4. Space & time: They are infinitely extended, which is of course an extensive magnitude. They are also infinitely divisible, which is an infinite cardinality: a region of space/time contains an infinite cardinal number of parts. There’s no intensive magnitude here.

My book reviews many more cases. I claim that the same core theory accounts for all or nearly all of the examples of possible and impossible infinities.

Qualifier: We can also distinguish these two kinds of magnitudes:

• Artificial magnitudes: These are magnitudes that, intuitively, are just “made up” by us, typically by performing some arbitrary mathematical operations on the real magnitudes that are given in nature. Ex.: you can define a quantity that is equal to 1 divided by 3 minus your mass in kilograms. That’s artificial.

• Natural magnitudes: These are the real magnitudes that exist in nature, not the artificial ones that we make up. A good test of naturalness is whether the magnitude has a causal role or appears in the laws of nature. E.g., mass, energy density, temperature, length, duration.

It doesn’t matter whether some artificial magnitude is “infinite”. If we define an artificial magnitude, we can always construct rules so that it goes infinite in some cases, but who cares? What we care about are the natural magnitudes. So the theory above is only about impossible natural magnitudes.

3. Explanation

Why would only infinite intensive magnitudes be impossible, but the other infinities be possible?

The underlying explanation is this:

• There are no infinite numbers. I.e., there is no number larger than every natural number, or larger than every real number. Why?

  • A number is a definite quantity.

  • “Infinity” doesn’t pick out a definite quantity; rather, “infinite” means something like “without limit”. Consider that if you add 1 to infinity, you still have infinity. But a definite quantity cannot be unchanged when it is increased by 1.

  • So infinity isn’t a number.

• The notion of “infinite cardinality” can be explained without positing infinite numbers. To wit: A set has infinite cardinality when, for every natural number n, it contains a subset with at least n members. This does not posit any number larger than all the natural numbers.

• “Infinite extensive magnitude” can likewise be explained without such numbers. E.g., space is infinitely large if: for every real number r, there is a region of space with a volume of at least r liters. This doesn’t require any number larger than all the real numbers.

• Infinite intensive magnitudes could not be explained in a similar way. E.g., if we say a singularity has infinite energy density, we are actually saying that energy density is a single quantity that is larger than every real number. This is impossible since there are no numbers larger than all the real numbers.

That’s why there can’t be infinite intensive magnitudes.