All knowledge consists of propositions constructed of concepts by which all that is known is identified. Language is the primary tool of human knowledge, because all concepts are constructed of words and definitions and all propositions are constructed of concepts. But language is not the only tool of knowledge.
Language, mathematics, geometry, symbolic logic, and Boolean algebra are all methods of identification. They are all human inventions that have no metaphysical existence. Language consist of symbols which represent concepts including all the concepts used in other methods. The other methods frequently have their own unique symbols because they are more useful to those methods.
[NOTE: There is no logical argument against calling all of the knowledge methods language, so long as the fact they are unique, not just a variation of verbal language. I choose to distinguish verbal (spoken and written language) from mathematics, symbolic logic, and geometry, for example, because verbal language is based on concepts that can identify anything, while mathematics and logic are strictly limited to concepts identifying a specific class of existents.]
[NOTE: The methods described here are all epistemological methods.]
For example, mathematics uses symbols like the Arabic numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9; but these symbols can also be identified by the words, zero, one, two, three, four, five, six, seven, eight, and nine. Obviously, using words would be very awkward, and in complex equations and advanced methods (scientific notation for example) practically impossible.
The symbols used in all methods, from language to logic, are arbitrary. Arbitrary does not mean without meaning or purpose, it means they are not discovered, but invented. The letters of an alphabet, or the symbols used in any language, must be invented by someone, and they could be almost anything, but once invented and assigned to what they represent in a particular language, will always have that assigned function, (or changes in them will occur very slowly), whether it is a phonetic language where letters represent vocalizations, or pictographic language where the symbols represent concepts directly. It is one of the aspects of such methods that the use of symbols must be consistent or they loose their cognitive value.
There is no metaphysical relationship between the symbols of a language and the concepts they represent, which is why a language must be learned, since there is no way to 'discover' the relationships between a language's symbols and the concepts they represent. It is also the reason why almost any word in one language can be translated into a word in another language. The words are arbitrary and determined by the language, the meanings are determined by the facts of reality the concepts identify. The assigning of the meaning to a word (definition) is a function of the language, the method, determined by the principles of epistemology.
While the symbols used in any of the methods by which knowledge is held and used are arbitrary, how they are used is not arbitrary, and they can only be used as they are because their use conforms to the nature of reality. For example, the counting numbers work because they are assigned a particular order, and must always be used in that order. One can count any number of things so long as the count is always performed by assigning a number to each item counted in the order 1, 2, 3, 4, ... 'last number.' If the numbers were assigned willy nilly, so to speak, counting would not say anything about the number of things counted.
The symbols used by a method are arbitrary, the method of using those symbols by which true aspects of reality may be identified must be discovered and conform to the nature of reality.
The methods themselves are not aspects of the metaphysical, they are entirely epistemological. It is the nature of reality that makes those methods valid when they conform to that reality, but there is nothing about those methods that exists independently of the methods, that is, independently of human minds. There are no numbers, no measurements, no logic, and no geometry in any aspect of metaphysical reality. Methods are invented by human beings, the fact those methods work and how they work is discovered, not by direct observation of the metaphysical (the methods do not exist metaphysically), but by comparing human knowledge and thinking to the facts of reality. The discovery is always on the order, "if I perform this mental operation, I will know this fact about reality."
Is the human mind, then, mathematical in nature as Rand, Peikoff, Harriman, and Binswanger suggest? The human mind is metaphysical, it exists and has the nature it has whether anyone knows it or anything about it. [Please see the chapter, "Mind," for a complete description of the nature of the mind.]
Since mathematics is a human invention, a method developed by discovering a means of identifying certain aspects of reality, it is impossible that the mind, itself, would have such characteristics, because they only exists epistemologically. Attributing the epistemological to the metaphysical is reification.
The failure to explicitly differentiate between metaphysical reality and the epistemological methods by which that reality is consciously comprehended (identified and known) is a source of endless confusion not only in philosophy, but in the sciences and all other intellectual disciplines as well.
We can use language and mathematics to discover and identify the nature of that which exists, but reality is no more "mathematical" than it is "Chinese" (if that is the language one uses to learn) or "French" (if that is the language one uses) or linguistic, whatever one's language is. The attributes of reality that mathematics is used to identify and explain are real metaphysical attributes, but in themselves they do not have any mathematical attributes. To assign mathematics to the metaphysical, as though the metaphysical itself had mathematical attributes, is a kind of anthropomorphism or mysticism. It is identical in principle to attributing the teleological to nature.
[NOTE: Ayn Rand, Dr. Peikoff, Harriman, and Dr. Binswanger all make the mistake of attributing mathematical characteristics to the mind and Dr. Binswanger makes the additional mistake of attributing the teleological to biology and evolution.]