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There are several issues of writing style that are particularly relevant in mathematical writing. In the interest of clarity, sentences should not begin with a symbol. Do not write: Suppose that G is a group. G can be decomposed into cosets, as follows. Let H be the corresponding subgroup of G. H is then finite. Instead, write something like:
Definition There should be an exact definition, in mathematical terms; often in a Definition(s) section, for example: Let S and T be topological spaces, and let f be a function from S to T. Then f is called continuous if, for every open set O in T, the preimage f −1 (O) is an open set in S. Examples
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.
In applied fields the word "tight" is often used with the same meaning. [2] smooth Smoothness is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to analyticity, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion ...
Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. For example, the physicist Albert Einstein 's formula E = m c 2 {\displaystyle E=mc^{2}} is the quantitative representation in mathematical notation of mass–energy ...
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.
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Rather than characterize mathematics by deductive logic, intuitionism views mathematics as primarily about the construction of ideas in the mind: [9] The only possible foundation of mathematics must be sought in this construction under the obligation carefully to watch which constructions intuition allows and which not. [12] L. E. J. Brouwer 1907