Discovery

It is commonly supposed that a theorem in mathematics, once proven, stays proven and remains a theorem forever, apart from occasional mistakes. This turns out not always to be the case, although exceptions are fairly rare. The best-known examples are supposed proofs that Euclidean geometry is the only one possible, and that there can be no consistent arithmetic containing infinitesimals, and the period of confusion in mathematical logic and set theory from the time of the discovery of the great paradoxes to the creation of non-standard mathematics and model theory.

Another kind of problem has been much more common. The mathematical profession has often objected to the introduction of new notations (Arabic numerals), and objects into mathematics (zero, negative numbers, complex numbers, infinite sets, infinitesimals), or to particular methods of proof (non-constructive proofs in infinite sets theory, use of the Axiom of Choice and its equivalents).

Thomas Kuhn proposed a theory of scientific discovery in which a single contrary fact meant doom for a theory. This turns out not to be true for science in general, and for mathematics in particular, which is fortunate for us, considering how many paradoxes of logic and set theory there were early in the 20th century. A single verified, thoroughly tested fact may require the creation of a more general theory or a new branch of a science, but the old theory is usually kept on in the domain where it is adequate. Examples include celestial navigation (as though the Earth stands still), and the use of Newtonian mechanics where none of Special Relativity, General Relativity, and Quantum Mechanics are required.

In all of these cases, math and science advance fastest when practitioners get the wrong answers according to the prevailing theory.