Then, linguistics comes in:
The finding vindicates a proposal first made on linguistic grounds by Joseph Greenberg, the great classifier of the world’s languages. He asserted in 1987 that most languages spoken in North and South America were derived from the single mother tongue of the first settlers from Siberia, which he called Amerind. Two later waves, he surmised, brought speakers of Eskimo-Aleut and of Na-Dene, the language family spoken by the Apache and Navajo.
But many linguists who specialize in American languages derided Dr. Greenberg’s proposal, saying they saw no evidence for any single ancestral language like Amerind.Well, not just specialists in American languages but historical and comparative linguists around the world.
The piece then concludes with this, a quote from Andres Ruiz-Linares of University College London, one of the investigators:
“Many linguists put down Greenberg as rubbish and don’t believe his publications,” Dr. Ruiz-Linares said. But he considers his study a substantial vindication of Dr. Greenberg. “It’s striking that we have this correspondence between the genetics and the linguistics,” he said.So, actually, Joseph Greenberg — who was an important linguist, somebody who was a founder of the modern field of linguistic typology (image from here) — claimed that almost all languages of the Americas are genetically related, save for the Na-Dene and Eskimo-Aleut families. He proposed this using tragically, horribly flawed data and using methods that can most gently be described as extremely controversial and not accepted by any serious historical linguist that I know or know of. The basic method is 'Mass Comparison', where you just eyeball lots of data. You can get a sense of some of the discussion in the journal Language 64.591-615 in 1988, where Lyle Campbell wrote a review article on Greenberg's book Language in the Americas. But there's a huge literature dedicated simply to cataloging the vast numbers of errors in Greenberg's data and some work showing that the errors consistently skew things in favor of his views.
As far as I know — and I don't have all the stuff at hand, but I know the work pretty well — linguists didn't particularly challenge claims about three waves of migration. If we think about basic geography, you might posit parts of this based on just distribution of languages – Eskimo-Aleut languages are pretty close to Siberia and a lot of Na-Dene ones are not much farther away. Then there's the rest of the hemisphere and it's hundreds of languages which look like they belong to many different families. Don't really need linguistics or detailed linguistic evidence for that.
Linguists really focused on the supposed evidence for 'Amerind' as a so-called super-family including hundreds of languages that cannot be shown to be related using the classic tools of comparative linguistics, like the comparative method. Most of us who talked about this in public and in print specifically said that we weren't claiming that the languages weren't related, just that we don't have evidence showing it or methods that allow us to show it. There IS evidence emerging for connections between some languages of Siberia and the Na-Dene family (search this blog for Ed Vajda's name), and that could easily correlate with them having come in a distinct migration. But that's as much as we've got right now.
I can build a theory about how dry weather leads to plants dying that rests on gremlins killing plants because they're angry about the absence of rain, because they like to dance in the rain. Whether gremlins play a role is independent of whether dry weather correlates with plants dying. (Watching things here, though, I'm pretty confident about the correlation.) Greenberg's observation about the distinctiveness of Eskimo-Aleut and Na-Dene vis-à-vis other languages of the western hemisphere is an interesting one, as is the claim of three waves of migration which he connected to that. And the claim of waves and these particular waves might be right — the genetic evidence sounds like it's consistent with it and the geography would fit neatly — but the purported linguistic arguments are all about gremlins.
My point is that it's best to see Greenberg's views about waves of migration as a CLAIM that may or may not be right. You really cannot say that it's supported by linguistic evidence nor that 'Amerind' is supported by linguistic evidence.