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Concept
Language of mathematics
Summary
The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) with concision, precision and unambiguity.
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.
Use of neologisms. For example polynomial, homomorphism.
Use of symbols as words or phrases. For example, and are respectively read as " equals " and "for all ".
Use of 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.
Use of 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.
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.
H. B.
Official source
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Related courses (1)
PHYS-757: Axiomatic Quantum Field Theory
Presentation of Wightman's axiomatic framework to QFT as well as to the necessary mathematical objects to their understanding (Hilbert analysis, distributions, group representations,...).
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