The mathematician, logician and philosopher Gottlob Frege made an interesting statement about language in one of his articles, “Ueber die wissenschaftliche Berechtigung einer Begriffsschrift” (“About the Scientific Justification of a Concept Script”). After writing about the deficits of natural language, especially its ambiguity and vagueness, he states:
The highlighted shortcomings are due to a certain softness and mutability of language. On the other hand, it is precisely this softness and mutability that gives language its versatility and its ability to develop. In this respect, language can be compared to our hands’ ability to adapt themselves to diverse tasks. However, our hands are insufficient for us, so we created artificial hands—tools for special purposes—and these allow us to do the precise and detailed work which our hands would not be able to do.
The topic of the article is the justification of the introduction of a Frege’s “Begriffsschrift”, or concept script, a formal language for logics and mathematics. Frege compares it to a special purpose tool that allows the hand to do things it could not do alone and that enables it to work more exactly. What I find interesting here is the insight that exactly what makes language insufficient for the purposes of logics or mathematics gives it its adaptability and potential for development. Clearly, what Frege had in mind was not the project of a formal semantics for natural language, but an extension of language by an exact formal tool for special purposes. Later projects to describe the semantics of natural language in terms of formal logics sometime refer to Frege, who undoubtedly contributed some ideas towards such a project. But I don’t think he had such a project in mind himself.
Interestingly, Frege speaks of the ability to develop (“Entwicklungsfähigkeit” – this could also be translated as “potential for evolution”). The development of new formal extensions to language, like Frege’s own conceptual notation, is an example of such evolution.
My own opinion on this matter is that since language is extensible like this, a complete formal theory of language, of its syntax and semantics, is not possible. It seems to me that Frege had a similar view. Further down in the article we see that he had an evolutionary or developmental idea of how language, thinking and technology develop in a circular or spiral-like way. When discussing the possibility of developing an exact notation starting from the less exact natural language, he states:
Exactly the same difficulty arises in the case of language already. Language is supposed to have facilitated the development of reason but how could human beings create language without reason. In order to investigate the laws of nature, physical devices are used. These can only be created by an advanced technology which in turn is based on knowing the laws of nature. In each case, the circle is solved in the same way. A progress in physics causes a progress in technology and this in turn enables us to build devices which promote physics. The application to our case is obvious.
Frege created the foundations of modern formal logic and this in turn spawned the philosophical programs of logicism and of formal semantics for natural language.
But in this relatively little-known article Frege is pointing the way into another direction, looking at language (and thinking) as an artificial creation of humans that is versatile and developing, historic and unfinished, augmentable by formalisms but not based on them. Frege’s aim, I think, was not to provide a semantics of language in terms of logic, but to develop language and thinking further and improving their exactness. His later writings on language may be reinterpreted along these lines, not as an attempt to formalize semantics, as if an underlying logical structure of language had always been there, but to improve the exactness of speaking, writing and mathematical and philosophical thinking.
The later Wittgenstein’s comparison of language with an ancient city comes to mind here, a maze of little streets and squares, surrounded by new quarters with strait regular streets. Frege erected one of these new regular quarters, perhaps part of the university district of the city, but he did not aim at braking down the historical center and replace it with strait rectangular streets as well.
(The picture is from http://commons.wikimedia.org/wiki/File:Young_frege.jpg)
 „Ueber die wissenschaftliche Berechtigung einer Begriffsschrift“, Zeitschrift für Philosophie und philosophische Kritik, 81, 1882, p. 48 – 56 (republished in Ignacio Angelelli (ed.) „Gottlob Frege Begriffsschrift und Andere Aufsätze“, Darmstadt, 1964, p. 114). I have published the complete German text here and an English translation as:
Gottlob Frege: On the Scientific Justification of a Concept Script. (translated and edited by Andreas Keller and Tina Forsee), Edition Nannus, 2016 (version 2), URL = <https://editionnannus.wordpress.com/2016/10/09/gottlob-frege-on-the-scientific-justification-of-a-concept-script/>.
 In Wittgenstein’s „Philosophical Investigations“.