When I knew about Major Mnemonic System, I thought: "Are there the languages that have writing systems consisting only numbers?" Do they exist?
1) The Major Mnemonic system isn't a language. It's just a coding system, but considering that to be a language, the binary codes fed to a computer is also a language and so on. 2) A language's purpose is to be able to convey thoughts. Thus such a language (which has just numbers) would only exist in a society where people just talk about numbers (which is obviously not possible) , not even mathematicians can talk just using numbers, because they have a "human" life to live which requires words/sounds to convey that meaning. 3) (a personal opinion) Number words came into a language , only after nouns or "object" words. One would have counted something only after she knew what is to be counted. Thus showing that other words "needed" to exist before numbers. And that the numbers aren't enough.
(P.S. this answer is to an older version of the question)
I think you are using those terms a little loosely, and when you pin them down to the usual ideas they represent, that doesn't exactly make sense. But it does give a vague idea of what you're looking for.
A language is a communication device and, especially here on linguistics.SE, one between humans to communicate ideas. Then there are recording devices, like writing or codes, that map one thing into another which has certain properties, such as permanence and replicability, like writing, or efficiency and organization, like codes (Morse code, ASCII code, ).
Human languages have many possible recording technologies, one of which is writing. Usually these are linear codes that capture the pronunciation or meaning step by step over the course of an utterance. Human language communicate many concepts including numbers. If you had a writing system, you could possibly have writing elements that capture numbers, a small set of the concepts a human language wants to get across, but most of the elements of writing would probably represent better most other non-numerical concepts (like 'dog' or 'truth').
There are many writing systems for human languages and the ways they represent numbers is sometimes by having separate symbols for the numbers (that's the modern way), but in the past it is often by using existing symbols for other words or letters. But there are no natural writing systems that have symbols for numbers that are reused to represent phonology or concepts.
For example, Ancient Greek had no separate set of characters to represent numerals by themselves. It used the pronunciation letters, in its alphabetic ordering, to represent numerals as a secondary use. That is, alpha represented the digit that is written '1' in English, beta for '2', ... iota for '10' and so on.
So the basic Greek alphabet, used normally for pronunciation guide, could also be used for accounting/arithmetic.
But there is no natural writing system that starts off with numbers and these get reused for the rest of the natural language.