The parable of the pebbles

Decomposed granite in a bucket

Decomposed granite, courtesy bptakoma
How would you count the pebbles in this bucket?

Let’s look at one of the simplest rational methods, namely counting, to get clear how that works first. Later sections will extend this understanding to more complex cases.

An ancient shepherd, in an era before numbers were invented, is sick of his job. In the morning, he lets his sheep out of the pen to graze, and they scatter across a rugged pasturage. In the evening, he wants to be certain they have all returned to the fold, where they are guarded by sheepdogs against wolves. That takes a long walk, searching the hills and valleys in case a straggler has gotten stuck in a rocky crevice.

He discovers a way to divine whether some sheep are still out grazing, without searching. In the morning, as each sheep leaves one-by-one, he drops a pebble into a bucket by the gate. In the evening, as each returning sheep enters, he takes a pebble out of the bucket. When there are no pebbles left in the bucket, he can close the gate and turn in for the night.1

Why does this method of counting, using external physical counters, work? The usual rationalist story is that the number of pebbles in the bucket represents the number of sheep out to pasture. This representation is a correspondence, a “mirror of nature.”

This theory is true, as far as it goes. In fact, we can prove the shepherd’s method is correct, as an eternal fact about the deep structure of reality. Let us proceed by mathematical induction. At time t = 0, first thing in the morning, the bucket is empty and the gate is closed, so the number of sheep out to pasture s0 = 0, and the number of pebbles p0 = 0, and 0 = 0 so s0 = p0. At any time t when a sheep goes out of the gate, the number of sheep out increases by one, st = st − 1 + 1, and so does the number of pebbles, pt = pt − 1 + 1. By the inductive hypothesis, we have st − 1 = pt − 1. Substituting equals for equals, st = st − 1 + 1 = pt − 1 + 1 = pt. The case of a sheep coming in through the gate can be handled symmetrically, with a unit decrement rather than increment. So for all times t, pt = 0 ⟹ st = 0. QED.2

But… what about a bucket of pebbles makes them represent a field of sheep? Suppose someone from the next town happens to walk by—on her way to the fair—with a bucket of 37 pebbles. Suppose there happen to be 37 sheep in the pasture; does her bucket represent that? Presumably not. But suppose her 37 pebbles are physically identical to the shepherd’s. What non-physical property do his pebbles have that makes them representational, although hers aren’t?

Most rationalisms run into serious trouble here. They suppose that beliefs are some sort of thing in your head. What makes that sort of thing representational? What property of a collection of neurons makes them represent “Ludwig has fed the dog” or “all ravens are black”?

In the 1980s, this conundrum, “the problem of intentionality,” became the central issue in the philosophy of mind. Cognitivism collapsed when it became clear that no answer is possible—not because we don’t know enough details about how brains work, but even in principle.

It’s not that the shepherd’s bucket or pebbles are different from the fair-goer’s. It’s that he’s doing ongoing work to make them correspond with sheep, and she isn’t. Representation is not a property of the bucket, pebbles, or sheep. It’s a property of the whole history of interaction of the bucket, pebbles, shepherd, sheep, and gate. Likewise, beliefs aren’t in your head; they too are dynamics of interaction. Representation can’t be found in a snapshot of the state of the world, nor in a timeline of brain activity. It’s necessarily a process extended in both time and space.3

Rationality works because we make it work—not because it is eternally, absolutely, ultimately Correct.4 The meta-rationalist agenda asks: what kinds of work, specifically, do we do to make rationality work? How can we do that work better?

In the parable so far, the task appears perfectly mechanical and mindless. The shepherd simply puts one pebble in the bucket when a sheep goes out, and removes one when a sheep comes in. A rationalist account can accommodate error: it’s understood that if the shepherd somehow adds two pebbles by mistake, or forgets to remove one, the count will come out wrong. These are deviations from rationality. If rationalism is taken as a normative (not descriptive) theory, there is no problem here. Either the shepherd executes rationality, or not.

The method is rational because it’s mindless. A turnstile mechanism that drops pebbles in the bucket when a sheep passes could be more reliably rational than the shepherd, who is probably daydreaming about smooching with the fair-goer instead of doing his job.

However, the correctness proof depends on an ontology according to which pebbles, sheep, and containment are perfectly definite. There must be an absolute truth about how many pebbles are in the bucket. Buckets are about the same size as eggplants, and as we have seen, there are few absolute truths to be found at that scale.

What is a pebble? It’s a chunk of rock, not too big or small. How small is too small? There is no absolute truth about that. How many things are in the bucket? One person might say “six pebbles”; another might say “there’s one small stone plus five pebbles”; another “four pebbles, a couple little flakes, and some sand—I’m not going to count the grains! Bits of dust too, I suppose.” Who is right?

To make the method work, you must choose bits-of-rock of roughly even sizes, so you can distinguish them from littler bits—stray grains of sand or dust in the bucket—that don’t count. How even? Even enough that you can make a reliable-enough judgement. How even is that? This requires a judgement about your own ability to make the first judgement reliably enough.

Also, the pebbles you find in a sheep field are not definite objects. They are atoms that are more-or-less stuck together. How stuck-together is enough? Most pebbles where I live are decomposed granodiorite, which is highly friable. Often you can crush them between your fingers. If you drop one in a bucket, it may well break apart, and then there are two pebbles. Or maybe that’s one pebble plus a flake that doesn’t count, and some sand and dust? To prevent such confusion, you must choose pebbles that are unlikely to fracture.5 How can you tell? Perceptual skill born of experience playing with pebbles.

“How many pebbles are in the bucket?” isn’t a theoretical problem, because it doesn’t and can’t have a theoretical answer. It is a practical problem. Pebbles and sheep mostly behave like definite objects.6 You can’t count clouds or jam, because they mostly don’t. Counting depends on things reliably-enough remaining distinct-enough, or on your having a good-enough practical method for imposing individuation. There is no absolute truth about distinctness, so there’s no absolute truth about numbers of physical objects.

So the rational counting procedure cannot simply be executed mindlessly. It requires circumrational support. That is composed of merely-reasonable ongoing activities that make a rational ontology accurate enough that rational conclusions work well enough. (In this case, that is the ontology of discrete, countable objects.) Circumrationality generally involves non-rational tasks of perception, interpretation, improvisation, and negotiation. (“Using a rational framework,” below, explains this in more detail.)

Counting works only because we make it work.

Couldn’t we select pebbles rationally, by specifying technical criteria of size and cohesiveness, and testing those with objective measurements? Yes, but that wasn’t part of the shepherd’s method, which we supposedly proved correct. It’s a much bigger job, and one not feasible for someone who hasn’t even got numbers yet.

Systematization replaces merely-reasonable judgements with objective, rational criteria. It’s one kind of work we can do to make rationality work better. Whether or not it is a good move depends on purposes, circumstances, and resources. Systematization usually requires a large up-front investment, which may result in greater reliability and/or efficiency later.

The modern world was built on systematization and depends on its often working well. However, systematization is not always feasible, and can never be complete. It can expand the territory of rational operation, but a rational system always has margins where it interfaces with nebulous reality, so it always requires circumrational support to manage those margins. Failure to recognize this results in characteristic patterns of breakdown—discussed at the end of this Part—which can only be addressed meta-rationally.

  • 1. Versions of this parable are often used in discussions of counting in the philosophy of mathematics. I don’t know where it originated. The earliest I know of is John F. Lucas’ 1986 Introduction to Abstract Mathematics, p. 119. My thanks to Lucy Keer for pointing it out to me. Stephen Kleene’s 1952 Introduction to Metamathematics tells a somewhat similar tale, of tethering sheep to trees. Thanks to David MacIver for alerting me to this one. I first encountered the trope in Eliezer Yudkowsky’s “The Simple Truth,” which draws many, but not all, of the same conclusions I do.
  • 2. To some, this may seem a ridiculously detailed proof for an obvious proposition. If you are a logician, you might instead consider it a lot of vague hand-waving and hot air. If you want to be really sure, you can formalize the problem with the Presburger axioms for addition. The Peano axioms, which cover multiplication as well, are famously impossible to prove consistent, so you shouldn’t believe anything anyone says about the times table. 3 × 3 = 9? Highly speculative! But Presburger arithmetic is provably consistent, complete, and decidable. You can give your proof to a program which will check it and definitively declare it to be absolute eternal truth. Phew, what a relief!
  • 3. The pebble-counting analogy is a deliberately simple case. It should not be assumed to extend into a general theory of representation—mental or otherwise. It depends on a constant close causal coupling, whereas we can represent states of affairs from which we are currently causally isolated; or that are hypothetical, so that no causal coupling is possible at all.
  • 4. The math presumably is eternally, absolutely, ultimately Correct; but rationality is not just math, it’s the use of formal methods in the real world.
  • 5. Try looking in a nearby stream. Water-polished pebbles are better behaved than field pebbles. The “object” idealization is more nearly true of them.
  • 6. Sheep aren’t always well-behaved objects, either. Exactly when a pregnant mammal becomes two mammals is not an objective fact, but—in the United States—a matter of intense political debate. Setting that aside, pragmatically, if a ewe goes out fat and comes back thin, you should reasonably expect an extra lamb, and adjust your count accordingly. Still more problematically, from a philosophical point of view, incomplete embryonic separation can result in what could be considered either “Siamese” twins or a “monster” with two heads. Whether that should be counted as one sheep with two heads or two sheep that share body parts is not a well-defined question, and cannot be addressed rationally. Fortunately, this is not a practical problem. As a shepherd, you probably have at most one instance, so you can special-case it by making a reasonable judgement. If indeterminate cases were common, you’d have to come up with some more systematic way of categorizing them.

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This page is in the section In the cells of the eggplant,
      which is in ⚒ Fluid understanding: meta-rationality,
      which is in ⚒ Sailing the seas of meaningness,
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