Steven Pinker (in The Language Instinct, 1995) suggested a name for the language that people "think in": mentalese. This language, of course, is not a real language and it only vaguely resembles the actual languages that people speak or sign. On the other hand, it is universal; every human being is born with it. His argument is that no natural language could possibly serve as a good medium for reasoning, for several reasons: ambiguity, lack of logical explicitness, co-reference, deixis and synonymy.
In his own words:
People do not think in
English or Chinese or Apache; they think in a language of thought.
This language of thought probably looks a bit like all these languages;
presumably it has symbols for concepts, and arrangements of symbols
that correspond to who did what to whom, as in the paint-spraying
representation shown above. But compared with any given language,
mentalese must be richer in some ways and simpler in others. It
must be richer, for example, in that several concept symbols must
correspond to a given English word like stool or stud. [...]. On the other hand, mentalese must
be simpler than spoken languages; conversation-specific words and
constructions (like a and the) are absent, and information about
pronouncing words, or even ordering them, is unnecessary. Now, it
could be that English speakers think in some kind of simplified and
annotated quasi-English, with the design I have just described, and
that Apache speakers think in a simplified and annotated quasi-
Apache. But to get these languages of thought to subserve reasoning
properly, they would have to look much more like each other than
either one does to its spoken counterpart, and it is likely that they
are the same: a universal mentalese.
Knowing a language, then, is knowing how to translate mentalese
into strings of words and vice versa.
The same argument can, of course, be applied to deaf-mute people, even to those who have never learned any language at all. He presents a few compelling examples of people with serious disabilities who are nevertheless capable of the same kinds of complex reasoning that speaking and signing people are.
This is what cognitive psychologists and neuroscientists call "computational" or "representational" theory of mind, according to which, what takes place in the mind is a mere manipulation of symbols, much like a computer, hence the name. Thus, the so called mentalese would be a very complex system of symbols used to represent the world inside people's minds. The "computer processor", in this case, would the brain, storing and manipulating these symbols by means of physico-chemical processes.
To sum up: there is no reason (according to this theory) to posit a natural language that people use for their thoughts. Languages would play, at best, an auxiliary role in the reasoning process.