the lord of the rings - What language did Hobbits speak before Men?

In Appendix F of The Lord of the Rings we are told that there “is no record of any language peculiar to hobbits. In ancient days they seem always to have used the languages of Men near whom, or among whom, they lived.”

This is a singularly striking statement from an author who otherwise insists that his pattern of literary creation is to begin with an invented language and then proceed to create creatures who speak that language and a world in which it is spoken. So where the nature of most of the peoples and races of Middle-earth bears a direct relationship to their language, hobbits appear linguistically as a cipher. For so far back as any records or memories reach, hobbits have been chameleon-like creatures who have adopted the language of their immediate human neighbours.

(italics are mine) Source: http://www.tolkienlibrary.com/press/1130-on-the-origin-of-hobbits.php

As a student of Cultural Anthropology and Linguistics who studied during the 80’s I must believe that a culture cannot be separated from its language as defined by the second paradigm. Language defines a culture and its origins, and I believe that Tolkien would agree with that.

So what is the origin of the Hobbits?

Who created them?

The Prologue to The Lord of the Rings says this:

It is plain indeed that in spite of later estrangement Hobbits are relatives of ours: far nearer to us than Elves, or even than Dwarves. Of old they spoke the languages of Men, after their own fashion, and liked and disliked much the same things as Men did. But what exactly our relationship is can no longer be discovered. The beginning of Hobbits lies far back in the Elder Days that are now lost and forgotten. Only the Elves still preserve any records of that vanished time, and their traditions are concerned almost entirely with their own history, in which Men appear seldom and Hobbits are not mentioned at all. Yet it is clear that Hobbits had, in fact, lived quietly in Middle-earth for many long years before other folk became even aware of them. And the world being after all full of strange creatures beyond count, these little people seemed of very little importance.

(once again, italics are mine)

1) ….to begin with an invented language and then proceed to create creatures who speak that language…

2) ...The beginning of Hobbits lies far back in the Elder Days that are now lost and forgotten.

So if the origin of the Hobbits predates that of Men, and Tolkien’s method was to create a language and then a people to speak it, what language did the Hobbits speak as a common tongue before the advent of Men? And why is there not any remnant of this older tongue existing in the Hobbit's everyday speech?