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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 ...view middle of the document...

Some of Zeno's most interesting arguments were directed against philosophical beliefs connected with the mathematical theories of the Pythagoreans. In spite of their interest in mathematics, the Pythagoreans lacked an efficient system of numerals--they represented numbers by groups of dots similar to those found today on dominoes and dice. Use of this notation was probably connected with their failure to distinguish sharply between units of arithmetic, geometic points, and physical atoms. . . .

Another of Zeno's arguments, one that continues to receive attention even in our own time, is intended to prove that motion is impossible. As background to this argument, suppose a runner were to advance from a starting point S to some goal G. In order to attain G, he must reach the mid point between S and G, which we can call M. Then, in order to get from M to the goal G, he must reach the midpoint between M and G, which we can call N. But to get from N to G he must reach the midpoint of this segment, which can call O. This can be repeated endlessly. Therefore, we have an infinite sequence of intervals, each of which must be traversed for the runner to reach his goal. . . .

In this second argument, Zeno employs a strategy similar to that found in the first. He supposes the runner reaches the goal and shows that this leads to impossible consequences. From this he concludes that the runner could not reach the goal. But, since the runner can be taken to represent any object moving from one point to another, it follows that no motion is possible.

These arguments are similar in structure: each begins with a supposition from which we deduce an impossible consequence (in the first case a contradiction), we then conclude that the original premise is false. This is a common and powerful form of reasoning that is important in the writings of Zeno's successors; Plato...

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