We know today that primates can communicate in a much more sophisticated way than we ever imagined. The famous pygmy chimpanzee, Kanzi, is perhaps the best example. But, there is still a huge gap between the most sophisticated things we've seen chimpanzees doing and what every human child is doing. Obviously, that is something that involved long evolutionary and historical processes. But we have very little to go on to reconstruct this process, and, of course, there have been hundreds of proposed scenarios for how it would have happened.
A little later in the interview, there is this interesting exchange between Boulton and Deutscher:
Boulton: ...without needing to date it scientifically, and without bringing any kind of religious charge to this, it's interesting that there's this biblical notion of, "in the beginning there was the word"--that a connection is there, somewhere.
Deutscher: Right, I think we instinctively feel it, and it must also reflect, in some sense, a historical reality that will forever will be a mystery.
What can be said about all this? First of all, I should probably point out that Boulton's seems confused about what the phrase "in the beginning was the word" meant in its original context. The phrase is taken from the first first verse of the first chapter of the Gospel of John in the New Testament. In its context, "word" (logos in the original Greek) does not mean "a word"--that is, a unit of language. Rather, it means "the Word," that is, the Lord Jesus Christ, who "was with God" and who "is God."
Nevertheless, putting aside Boulton's misuse of the biblical phrase, there are at least two interesting things to note here. First of all, Deutscher acknowledges that there is a "huge gap between the most sophisticated things we've seen chimpanzees doing and what every human child is doing." In other words, the human capacity for language is far beyond anything any animal, no matter how intelligent, can produce. Thus, explaining how human language could have evolved from the means of communication employed by animals will inevitably be a difficult task. In fact, according to Deutscher, it "will forever be a mystery." Second, despite our inability to explain how human language could have evolved, Deutscher takes it for granted that it did ("Obviously, that is something that involved long evolutionary and historical processes"). However, we should ask: on what grounds should he believe this? He doesn't say, but I suspect it is because Deutscher is committed to a naturalistic explanation for the origin of language. This, however, is a matter of philosophical commitment, not scientific fact.
All of this leads to a further problem. Deutscher's book, as I understand it, purports to explain the evolution of language, and yet he is unable to explain how human language came into existence. He just assumes that somehow it did and then proceeds to explain how language evolved from its most primitive form. This is analogous to evolutionary biologists telling us that they don't know how the first form of life emerged. Nevertheless, they just assume that it somehow it did and then attempt to explain how more complex forms of life evolved from that most primitive form of life (without explaining where that came from). It is as if the question of how language (or life) first sprang into existence is not really that important. And yet, it seems to me that the whole credibility of an evolutionary explanation for language (or life) rests initially on the plausibility of the idea that it "just happened" by itself, without the involvement of something--or Someone--outside of Nature.
In short, without meaning to, in his interview, Deutscher demonstrates at least two problems with an evolutionary explanation for human language: first of all, the "huge gap" between animal communication and human language, and second (and related to the first), the inability of linguists to provide a plausible naturalistic explanation of how language came into being. These should make us quite suspicious of any effort, like Deutscher's--no matter how sophisticated--to explain how human language could have emerged without the involvement of a Creator.
Image of Guy Deutscher from Wikipedia