The unknowability part of epistemicism about vagueness

One way to present epistemicism is to say that

1. vague concepts have precise boundaries, but

2. it is not possible for us to know these boundaries.

A theist should be suspicious of epistemicism thus formulated. For if there are precise boundaries, God knows them. And if God knows them, he can reveal them to us. So it is at least metaphysically possible for us to know them.

Perhaps the “possible” in (b) should be read as something stronger than metaphysical possibility. But whatever the modality in (b) is, it seems to imply:

3. none of us will ever know these boundaries.

But if epistemicism entails (c), then we don’t know epistemicism to be true. For if there are sharp boundaries, for all we know God will one day reveal them to a pious philosopher who prays really hard for an answer.

I think the best move would be to replace (b) with:

4. it is not possible for us to know these boundaries without reliance on the supernatural.

This is more plausible, but it seems hard to be all that confident about (d). Maybe there is some really elegant semantic theory that has yet to be discovered that yields the boundaries. Or maybe our mind has natural powers beyond those we know.

Let me try, however, to offer a bit of an argument for (d). Let’s imagine what the boundary between bald and non-bald would be like. As a first attempt, we might think it’s like this:

1. Necessarily, x is bald iff x has fewer than n hairs.

But there is no n for which (1) is true. For n would have to be at least two, since it is possible to be bald but have a hair. Now imagine Bill the Bald who has n − 1 hairs, and now imagine that these hairs grow in length until each one is so long that Bill can visibly and fully cover his scalp with them. At that point, Bill wouldn’t be bald, yet he would still have n − 1 hairs. So, the baldness boundary cannot be expressed numerically in terms of the number of hairs.

As a second attempt, we might hope for a total-length criterion.

2. Necessarily, x is bald iff the total length of x’s hairs is less than x centimeters.

But it is possible to have two people with the same total length of hairs, one of whom is bald and the other is not. For the thickness of hairs counts: if one just barely has the requisite total length but freakishly thin hairs, that won’t do. On the other hand, clearly x would have to be at least four centimeters, since a single ordinary hair of four centimeters is not enough to render one non-bald, but one could have a total hair length of four centimeters and yet be non-bald, if one has four hairs, each one centimeter long and 10 centimeters in diameter, covering one’s scalp with a thick keratinous layer.

So, we really should be measuring total volume, not length. But there are other problems. Shape probably matters. Suppose Helga has a single hair, of normal diameter, but it is freakishly rigid and long, long enough to provide the requisite volume, but immovably sticking up away from the scalp and providing no coverage. Moreover, whatever we are measuring has to be relative to the size of the scalp. A baby needs less hair to cease to be bald than an adult. But it’s not just relative to the size of the scalp, but also the shape of the scalp. If one has a very large surface area of scalp but that is solely due to many tiny wrinkles, one doesn’t need an amount of hair proportional to that large surface area. To a first approximation, what matters is the surface area of the upper part of the convex hull of one’s scalp. But even that’s not right if we imagine a scalp that has very large wrinkles.

So, in fact, we have good reason to think the real boundary wouldn’t be simply numerical. It would involve some function of hair shape, volume and rigidity, as well as of scalp shape and size. And if we think about cases, we realize that it will be a very complex function, and we are nowhere close to being able to state the function. Moreover, to be honest, there are likely to be other variables that matter.

At this point, we start to see the immense amount of complexity that would be involved in any plausible statement of the precise boundary of baldness, and that gives us positive reason to doubt that short of something supernatural we could know where the boundary lies.

But suppose our confidence has not yet been quashed. We still have other serious problems. What we are looking for is a perfectly precise necessary and sufficient condition for someone to be bald. In that definition, we cannot use other vague terms. That would be cheating. What the epistemicist meant by saying that we don’t know where the boundaries lie was that we do not know any transparently precise statements of the boundaries, statements not involving other vague terms. But “hair” itself is a vague term. Both hair and horns are made of keratin. Where does the boundary between hair and horns lie? Similarly, “scalp” is vague, too. And it’s only the volume of the part of the hairs sticking out of the scalp that counts—the size of the root is irrelevant. But “sticking out” is vague, as is obvious when we Google for microscopic photography of scalps. And which particles are in the hair or in the scalp is going to be vague. Next, any volume and surface area measurements suffer from vagueness even if we fix the particles, because for quantum reasons particles will have spread out wavefunctions. And then Relativity Theory comes in: volume and surface area depend on reference frame, and so we need a fully precise definition of the relevant reference frame.

Once we see all the complexity needed in giving a transparently precise statement of the boundary of baldness, it becomes very plausible that we can’t know it by natural means, just as it is very plausible that no human can know the first million digits of π by natural means.

And things get even worse. For humans are not the only things that can be bald. Klingons can be bald, too. Probably, though, only humanoid things are bald in the same sense of the word, but even when restricted to humanoid things, a precise statement of the boundary of baldness will have to apply to beings from an infinite number of possible species. And the norms of baldness will clearly be species-relative. Not to mention the difficulty of defining what hair and scalp are, once we are dealing with beings whose biochemistry is different from ours. It is now starting to look like a transparently precise statement of the boundary of baldness might actually have infinite complexity.