As I understand it, the standard answer is that you can't prove such a claim with the methods that linguists use.
Or: once you've proved it, so what? We already know that modern humans all have the same cognitive and perceptual capacities, and are only disputing the relative modularity and context dependence of those faculties.
This paper's premises are intriguing, if far from obvious, and its results are pretty compelling ,,, However, I have some concerns about what lies in between the assumptions and the results, especially concerning the way "Total Phoneme Diversity" is estimated.
Many such features are "areal" — spread over geographical areas of related (and even unrelated) languages, and also spread over local descriptive practices among linguists. The areas in question often approach continental size, and thus these phenomena (and our choices about how to code them) can have a big influence on the outcome of algorithms that look for world-wide clines in the "diversity" measures that result from these choices.
How large could any single population of humans speaking the same language have been prior to 10K BP? If we are very generous and assume that the distribution of languages was similar to today ... then maybe 100K speakers.
But who seriously believes that there were any groups of 100K speakers of a single language at such an early date? Prior to 10K BP, humans lived in hunter-gatherer societies, not the larger settled communities that formed after agriculture, let alone what one saw after the foundation of cities. For such societies very small groups are the norm. Would we expect to see any groups that had populations of more than a few hundred or a few thousand? ... if the norm for most languages was indeed a few thousand speakers at a maximum, then all languages in the earliest periods would be tiny.
Another issue is that Atkinson's model assumes that we are talking about loss of inventory when populations get smaller ...
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