Why not? Look at it from the bottom up:
Communication is a fundament of life, from intra-cellular to inter-cellular to inter-organism interactions (another fundament is the ability to keep oneself in a low entropy state, at the expense of the rest of the world).
Human speech is an evolution of mammal communication. It grew up in complexity, from grunts and other basic noises, along with our way of living, up to what we have now.
> And how do we make completely novel utterances that attempt to express our otherwise abstract thoughts?
Speech is a big collage. New is either the result of
* a recombination of the sub-parts of past speech
* the definition of a new word in terms of older words, or sometimes arbitrarily (for proper nouns).
Nothing fancy AFAICT.
This view is just to simplistic to hold its weight when you really look at the intricacies of language and its evolutionary history which, by the way, I would suggest comes from manual gesture and not grunting.
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But you're probably right about gestures.
Wild chimps have a vocabulary of about 66 signs. We can also observe tribes with languages more primitive than ours (no pronouns, for example). But there's a missing link of several millions of years of evolution between both.
What are the (known) intricacies of the evolution of our ability to communicate?
There's no definitive proof for the statistical argument, but a growing amount of (neuro)scientific evidence points to it. What's (are) your alternative hypothese(s)?
I think that most people who believe in some form of the motor theory of speech perception will also believe that speech evolved from manual gesture.
Others scoff at the motor theory. In fact, I'd say I'm in the minority by bringing it up with any regularity.
If the question, what is "known" about the evolution of our ability to communicate, I wouldn't have much to point you towards. Most is theory based on modern evidence, somewhat like armchair psychology. Other people point to our ability to integrate non-verbal gestures into our comprehension, activation of our motor cortex prior to semantic/phonetic network activation when disambiguating difficult speech sounds, our ability to synthesize visual/auditory sources of information when the visual information relates to speech gestures (mouth/tongue movements), etc.
Aren't these issues completely orthogonal?
That criticism can be lobbed at all abilities that we claim came about due to evolution - which, to be clear, is all of them. The statistical model would be the mechanism, but it wouldn't be the reason why it evolved. That answer is relatively boring, and is the same one as all evolutionary processes: it appeared randomly from mutation, and it provided benefit to those that had it.
Not just boring, but a totally banal and useless answer.
What kinds of mutations? In what sequence? How did it provide a survival benefit? What were the earlier forms of language like, and how did they become the languages spoken today?
Just saying "evolution did it" is about as informative as saying "God did it."
It's the domain specific tools that are up in the air, though, and what I'm interested in.