Once upon a time—around 600 B.C.—there was a man named Thales, who invented what we call “science.”


Rational Number Traditional Logic Conditional Form Primitive Term Logical Paradox 
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  1. 1.
    Babylonia and Egypt. In Thales’ time both civilizations had long mathematical traditions, and had amassed considerable geometric and arithmetic knowledge. While the extent to which this mathematics was already abstract or deductive when the Greeks acquired it is a matter of debate, it was certainly not organized into deductive systems such as the Greeks would shortly construct.Google Scholar
  2. 9.
    Pattern for a Material Axiomatic System. From Eves, A Survey of Geometry (Allyn and Bacon, 1972), p. 11Google Scholar
  3. 11.
    wiseacre. The Tortoise refers to Zeno of Elea (flourished c. 450 B.C.), author of a number of famous paradoxes that bear his name, the implications of which are still debated. One of these, the Paradox of Achilles and the Tortoise, runs as follows (from Ettore Carruccio, Mathematics and Logic in History and in Contemporary Thought (Faber and Faber, 1964), pp. 33–34. Reprinted with permission.)Google Scholar
  4. 12.
    Proofs. Some of this article is adapted from Greenberg, Euclidean and Non-Euclidean Geometries (Freeman, 1980), pp. 32–34.Google Scholar
  5. 13.
    G. H. Hardy. In the 1920s Hardy was, by his own estimate, the fifth best pure mathematician in the world. In later life, his power to create new mathematics waning, he turned to writing first-rate textbooks and the wonderful A Mathematician’s Apology, a defense of his life’s work, largely on aesthetic grounds (Cambridge University Press, 1940, 1967). Of the Apology novelist and playwright Graham Greene wrote in a review, I know no writing—except perhaps Henry James’s introductory essays—that conveys so clearly and with such an absence of fuss the excitement of the creative artist. Since 1967, reprintings of A Mathematician’s Apology have included a touching foreword by Hardy’s longtime friend, author and physicist C. P. Snow.Google Scholar
  6. 14.
    Example of a Material Axiomatic System. Adapted from Eves, op. cit., p. 15.Google Scholar

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