First off, mathematics is not science. Science is constrained in that it must agree with reality. Mathematics isn't constrained in that way. The hypothesis that space and time are continuous is, at least conceptually, a falsifiable hypothesis. Suppose that some future scientific experiment shows this conjecture to be false at some very tiny scale. Will this invalidate the mathematics of the real numbers? No. It won't. It will merely mean that our use of the reals to describe the universe is not quite correct.It is mandatory for any scientific system/theory to "clearly define its terms". If we do not do that problems arise
Secondly, mathematicians do describe their terms, very precisely -- in the form of undefined terms and axioms. The natural numbers are described by a certain collection of axioms, the integers by another collection of axioms, and the reals by yet another collection of axioms, and so on.
Lay people and scientists typically use the term "number" to mean the a member of the reals. Unless they mean something else. Mathematicians are precise and use terms such as integer, rational number, real number, complex number, p-adic number, quaternion, etc. There is no one concept in mathematics that definitively constitutes "number".