The Historical Roots of Logic

The Historical Roots of Logic

People have always reasoned. Even early civilizations made technical and intellectual advances that presupposed logical inferences. For example, the Babylonians discovered and used a theorem (now called the Pythagorean theorem) that expresses the relation between the sides and the hypotenuse of right triangles, and the Egyptians used the formula for calculating the volume of a rectangular pyramid 'h(a^2 +ab + b^2)/3'. Such results cannot be obtained without reasoning. However, neither civilization sought formal proofs for mathematical insights; formulas were regarded only as recipes or rules of thumb that could be followed to calculate desired numerical values. In these early cultures, there was no interest in proofs or arguments and so there was no interest in general criteria for the correctness of arguments.

Pythagoras (ca. 532 B.C.), a Greek, was the first historical figure known to have been interested in the construction of mathematical proofs. As a young man, Pythagoras traveled widely; he studied firsthand the mathematical results of the Egyptians and Babylonians. Later, he established a semireligious brotherhood whose members were concerned primarily with discovering and proving mathematical theorems. Members of the brotherhood probably were the first to prove the Pythagorean theorem, and they also may have discovered the first proof that the square root of two is irrational.

The Pythagoreans were not the only Greeks who were passionately interested in arguments and proofs. The early Greek philosophers are generally distinguished from the sages, prophets, and oracles of other early Middle Eastern civilizations precisely in that they sought to support their opinions with reasons. An outstanding early example was Zeno of Elea (ca. 490-430 B.C.), who constructed numerous brilliant arguments against the philosophical doctrines of his opponents. Some of Zeno's most interesting arguments were directed against philosophical...