Noam Chomsky in Inference:
In the Discourse on Method, Part V, Descartes argued that the creative use of language marked the distinction between human beings and other animals, and between human beings and machines. A machine may be impelled to act in a certain way, but it cannot be inclined; with human beings, it is often the reverse.5 Explaining why this is so, is the Galilean challenge.
In the modern era, the challenge, although occasionally expressed, was also widely ignored. Wilhelm von Humboldt is an especially suggestive case to the contrary:
The processes of language must provide for the possibility of producing an undefinable set of phenomena, defined by the conditions imposed upon it by thought. … It must, therefore, make infinite use of finite means… [emphasis added]
The capacity for language is species specific, something shared by humans and unique to them. It is the most striking feature of this curious organism, and a foundation for its remarkable achievements. This is in its full generality the Galilean challenge. The challenge is very real, and should, I think, be recognized as one of the deepest questions in the rich two-thousand-five-hundred-year history of linguistic thought.
Until the twentieth century, there was never much to say about the Galilean challenge beyond a few phrases. There is a good reason why inquiry languished. Intellectual tools were not available for formulating the problem in a way clear enough to be seriously addressed. That changed thanks to the work of Alonzo Church, Kurt Gödel, Emil Post, and Alan Turing, who established the general theory of computability. Their work demonstrated how a finite object like the brain could generate an infinite variety of expressions. It became possible, for the first time, to address part of the Galilean challenge directly, even though the earlier history remained unknown.
With these intellectual tools available, it becomes possible to formulate what we may call the Basic Property of human language. The language faculty of the human brain provides the means to construct a digitally infinite array of structured expressions; each is semantically interpreted as expressing a thought, and each can be externalized by some sensory modality, such as speech.