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Whenever mathematicians pioneer into the frontiers of thought, they often bring back little pieces of newly found mathematical knowledge, called theorems. Some of these have been found for so long that they have become stale and common (like the Fundamental Theorem of Algebra), while others are very new and strange (Arctic Circle Theorem???). But all of them state some sort of mathematical concept.
But of course, mathematics would be arbitrary and confusing if we didn't know why everything worked. So mathematicians create proofs, which build up on existing concepts and theorems, that show why it must be logically inevitable for the theorem to be true.