Cultural arbitrariness is not the thing that is at the root of how race doesn't exist.

On the old Radiolab episode about race, the producers used an interesting fact to make an argument that race doesn’t exist — that it’s entirely a social construct. It turns out that the genetic variability within races is greater than the variability between races; the average difference between two people of the same race is greater than that of groups of people across races. In that sense, the idea of race is not really meaningful. But the same is true for the sounds p and b.
Put your finger to your throat and say “ppuh.” Then say “bbuh.” The vibration you felt for the second one is called voicing; it’s supposed to be the only difference between p and b. That said, things get fuzzy fast. Say “pee.” “Pee” doesn’t start out voiced, but it ends that way (in contrast with “bee,” which is voiced more from the beginning). Depending on context, you can actually move voicing up a lot more and still be perceived as uttering a p. And you can move voicing down from the beginning and still be a b. There are big individual differences too, so that the thing that came out of my mouth as a p might have come out of your sounding like a b. In real everyday language, the fluctuations are so wild that the variability within p or b is greater that then variability between them.
Does race exist? As much as p and b do. So wait: Do p and b exist? It turns out that there are sharp people working to destroy the ideas of the sounds p and b. For example, cognitive scientist Bob Port put his career behind undermining the static approaches to phonology that permitted the idea of linguistic atoms. And there’s something to it. It turns out that p and b are really complicated. But he can still pronounce his name. It seems you don’t have to be able to draw a clear line between them for them to be used by reasonable people as ideas. To take them too seriously is wrong, and to think that they can’t be used responsibly, or even usefully, is also wrong.
p, b, and race all look superficially like basic building blocks, but really they are each a complicated result of things like physiology, culture, and the context of each instant. So they are constructs, but not just social constructs. Their cultural arbitrariness is not the thing that is at the root of how they don’t exist. What does it mean for you? These constructs aren’t insubstantial because they are nothing, they are insubstantial because they are complicated.

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