Einstein's Theories of Relativity and Gravitation by Albert Einstein, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. Axioms Made to Order
Axioms Made to Order
This sounds innocent enough; but in no way was Euclid able to devise a proof—or, for that matter, a disproof. So he took the only way out, and said that if the lines were parallel, obviously they extended in the same direction and made the angles equal. The thing was so obvious, he argued, that [129]it was really an axiom and he didn’t have to prove it; so he stated it as an axiom and proceeded. He didn’t state it in precisely the form I have used; he apparently cast about for the form in which it would appear most obvious, and found a statement that suited him better than this one, and that comes to the same thing. This statement tells us that if the transversal makes two corresponding angles unequal, the lines that it cuts are not parallel and do meet if sufficiently prolonged. But wisely enough, he did not transplant this axiom, once he had arrived at it, to the beginning of the book where the other axioms were grouped; he left it right where it was, following the proposition that if the angles were equal the lines were parallel. This of course was so that it might appeal back, for its claim to obviousness, to its demonstrated converse of the proposition.
Euclid must have been dissatisfied with this cutting of the Gordian knot; his successors were acutely so. For twenty centuries the parallel axiom was regarded as the one blemish in an otherwise perfect work; every respectable mathematician had his shot at removing the defect by “proving” the objectionable axiom. The procedure was always the same: expunge the parallel axiom, in its place write another more or less “obvious” assumption, and from this derive the parallel statement more or less directly. Thus if we may assume that the sum of the angles of a triangle is always exactly 180 degrees, or that there can be drawn only one line through a given point parallel to a given line, we can prove Euclid’s axiom. Sometimes the substitute assumption [130]was openly made and stated, as in the two instances cited; as often it was admitted into the demonstration implicitly, as when it is quietly assumed that we can draw a triangle similar to any given triangle and with area as great as we please, or when parallels are “defined” as everywhere equidistant. But such “proofs” never satisfied anyone other than the man who made them; the search went merrily on for a valid “proof” that should not in substance assume the thing to be proved.
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This book is part of the public domain. Albert Einstein (2020). Einstein's Theories of Relativity and Gravitation. Urbana, Illinois: Project Gutenberg. Retrieved October 2022.
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