J.N. Coldstream's A Protogeometric Nature Goddess from Knossos PDF

By J.N. Coldstream

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Previously proved). "By definition. . " (a previous step in the argument). "By rule . . ) A justijication may involve several of the above types. Having described proofs, it would be nice to be able to tell you how to find or construct them. Yet that is the mystery of doing mathematics: Certain techniques for proving theorems are learned by experience, by imitating what others have done. But there is no rote method for proving or disproving every statement in mathematics. (The nonexistence of such a rote method is, when stated precisely, a deep theorem in mathematical logic and is the reason why computers will never put mathematicians out of business-see DeLong, 1970, Chapter 5).

Usually, common sense will quickly give you the negation. If not, follow the above rules. Let's work out the denial of Euclid's first postulate. This postulate is a statement about all pairs of points P and Q; negating it would mean, according to Rule 6, asserting the existence of points P and Q that do not satisfy the postulate. Postulate I involves a conjunction, asserting that P and Q lie on some line I and that lis unique. In order to deny this conjunction, we follow Rule 5. " If we return to the example of the surface of the earth, thinking of a "line" as a great circle, we see that there do exist such points P and Q-namely, take P to be the north pole and Q the south pole.

23). 23 Q (b) Given a segment PQ and a ray AB. Construct the point C on AB such that PQ == AC. ) Exercise (b) shows that you can transfer segments with a collapsible compass and a straightedge, so you can carry out all constructions as if your compass did not collapse. 3. The straightedge you used in the previous exercises was supposed to be unruled (if it did have marks on it, you weren't supposed to use them). Now, however, let us mark two points on the straightedge so as to mark off a certain distance d.

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A Protogeometric Nature Goddess from Knossos by J.N. Coldstream

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