Mathematical rigor is in the axiomatic approach that pervades each statement of a theory together with its proof.
This allows to realy have a set of useful knowledge about a certain kind of mathematical phenomena.
A mathematical theory is a set of principles and consequences of the principles which allows to organize the knowledge in a systematic fashion. One of the first examples in the human culture is the euclidean geometry.
A useful science is characterized by its irrestrictable obedience of the rules of logic, not of ethics unfortunately.
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