**Differences With Scientists**

Mathematics differs from natural science in that scientists subject truth claims to tests by experiments, while mathematical propositions are conclusions of mathematical proofs.

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a *conjecture*, as opposed to an ultimate goal: a *theorem* that has been proved.

Scientific theories change when new information about the world is discovered. Mathematics changes in a similar way: New ideas do not falsify old ones, but new concepts refine old concepts and old theories, attaining a fuller understanding of the truth. One method of refinement is generalization, for example widening the scope of a concept. For instance, calculus (in one variable) generalizes to multivariable calculus, which generalizes to analysis on manifolds. The development of algebraic geometry from its classical to modern forms is a particularly striking example of the way an area of mathematics can change radically in its viewpoint, without making what was correctly proved before in any way incorrect; of course mathematical progress clarifies gaps in previous proofs, often by exposing hidden assumptions, which progress has revealed worth conceptualizing.

A theorem is true, and was true before we knew it and will be true after humans are extinct. Of course, our understanding of what the theorem *really means* gains in profundity as the mathematics around the theorem grows. A mathematician feels that a theorem is better understood when it can be extended to apply in a broader setting than previously known. For instance, Fermat's little theorem for the nonzero integers modulo a prime generalizes to Euler's theorem for the invertible numbers modulo any nonzero integer, which generalizes to Lagrange's theorem for finite groups.

Read more about this topic: Mathematician

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