Rigour in mathematics

Mathematics is very rigorous and everything must be proven properly even things that may seem true and obvious.

Can you give me examples of conjectures/theories that seemed true but through rigorous mathematical proving it was shown otherwise?

Answer

Finding the roots of a linear polynomial is trivial. Already the Babylonians could find roots of quadratic polynomials. Methods to solve cubic polynomials and fourth-degree polynomials were discovered in the sixteenth century, all using radicals (i.e. nth roots for some n).
Isn’t it obvious that finding the roots of higher degree polynomials is also possible using radicals and that we have not found the formulas yet is only because they become more and more complicated with higher polynomial degrees?

Galois theory shattered this belief.

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