"One cannot really argue with a mathematical theorem"

Mathematical theorems hold a special status in human knowledge due to their reliance on rigorous logic and precise definitions. When a statement has been proven as a theorem, it means that, based on the agreed-upon assumptions and axioms, its truth inevitably follows through deduction. The role of emotion, rhetoric, or persuasion vanishes in the face of mathematical proof. Unlike the debates common to philosophy, politics, or the arts, mathematics is defined by objectivity; once a theorem has been demonstrated, its correctness is beyond subjective dispute as long as the underlying logic and definitions are accepted.

Stephen Hawking’s observation points toward the unique certainty with which mathematics operates. When theorems are established, they become firm pillars on which subsequent knowledge is built. They offer a universality and permanence seldom found in other domains. Thus, to ‘argue’ with a mathematical theorem is, in effect, to dispute the underlying structure and agreement that mathematics is founded upon. Disagreement may occur about the relevance or application of a theorem to the world outside mathematics, or about the choice of axioms themselves, but within the system of mathematics, logical proof stands supreme. If a mistake exists within a proof, it can be identified and corrected, but if the proof is correct, debate is futile.

This clarity is a key reason for mathematics’ foundational role in the sciences. Through mathematical theorems, predictions can be made, structures analyzed, and phenomena explained with a precision unmatched by verbal reasoning alone. While human error in calculation or misunderstanding of application can occur, theorems themselves, once proven, possess a kind of inviolability. The ability to transcend subjective opinion grants mathematics an authority and dependability that continues to shape our understanding of the universe and the structures within it.