The main features of the mathematical language are the following.
Use of common words with a derived meaning, generally more specific and more precise. For example, "
or" means "one, the other or both", while, in common language, "both" is sometimes included and sometimes not. Also, a "
line" is straight and has zero width.
Use of common words with a meaning that is completely different from their common meaning. For example, a mathematical
ring is not related to any other meaning of "ring".
Real numbers and
imaginary numbers are two sorts of numbers, none being more real or more imaginary than the others.
symbols as words or phrases. For example, and are respectively read as " equals " and "for all ".
formulas as part of sentences. For example: "represents quantitatively the
mass–energy equivalence." A formula that is not included in a sentence is generally meaningless, since the meaning of the symbols may depend on the context: in " ", this is the context that specifies that e is the
energy of a
physical body, m is its
mass, and c is the
speed of light.
mathematical jargon that consists of phrases that are used for informal explanations or shorthands. For example, "killing" is often used in place of "replacing with zero", and this led to the use of assassinator and annihilator as technical words.
Understanding mathematical text
The consequence of these features is that a mathematical text is generally not understandable without some prerequisite knowledge. For example the sentence "a
free module is a
module that has a
basis" is perfectly correct, although it appears only as a grammatically correct nonsense, when one does not know the definitions of basis, module, and free module.
Now mathematics is both a body of truth and a special language, a language more carefully defined and more highly abstracted than our ordinary medium of thought and expression. Also it differs from ordinary languages in this important particular: it is subject to rules of manipulation. Once a statement is cast into mathematical form it may be manipulated in accordance with these rules and every configuration of the symbols will represent facts in harmony with and dependent on those contained in the original statement. Now this comes very close to what we conceive the action of the brain structures to be in performing intellectual acts with the symbols of ordinary language. In a sense, therefore, the mathematician has been able to perfect a device through which a part of the labor of logical thought is carried on outside the
central nervous system with only that supervision which is requisite to manipulate the symbols in accordance with the rules.: 291