数学上，严谨不同于生活中的严谨，它指数学系统（尤指公理系统）的完备性和兼容性。

完备性指公理数量不多不少正好可以推导出这门学科的全部结论；自洽性指公理系统内不存在悖论（即既是真又是假的命题）。比如绝对几何学加上第五公设就成为欧式几何，或者加上第五公设的反命题就成为非欧几何，但后两者并不满足完备性要求，只有绝对几何学才是度量几何类中的完备系统。自洽性与哥德尔不完备定理并不矛盾，前者断言不存在既真又假的命题，而后者断言存在既不可证明又不可证伪的命题，就好比第五公设之于度量几何，连续统假设之于集合论，选择公理之于ZF系统。

Rigour (BrE) or rigor (AmE) (see spelling differences) describes a condition of stiffness or strictness. Rigour frequently refers to a process of adhering absolutely to certain constraints, or the practice of maintaining strict consistency with certain predefined parameters. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mathematical proofs which must maintain consistent answers; or socially imposed, such as the process of defining ethics and law.