A completely rigorous treatment of mathematics, it might seem, would require us to define every term and prove every statement we encounter. However, unless we want to engage in circular reasoning, or have an argument that goes backwards infinitely far, we have to choose some place as a logical starting point, and then do everything else on the basis of this starting point. This approach is precisely what Euclid attempted to do for geometry in “The Elements,” where certain axioms were formulated, and everything else was deduced from them. (We say “attempted” because there are some logical gaps in “The Elements,” starting with the proof of the very first proposition in Book I. Fortunately, these gaps can be fixed by using a more complete collection of axioms, such as the one proposed by Hilbert in 1899, which made Euclidean geometry into the rigorous system that most people believed it was all along. The discovery of non-Euclidean geometry is a separate matter. See [WW98] for details on both these issues. This critique of Euclid, it should be stressed, is in no way intended to deny the overwhelming importance of his work.)
KeywordsMaximal Element Proper Subset Venn Diagram Logical Contradiction Logical Starting Point
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