# Breakthroughs in Mathematics by Peter Wolff PDF

By Peter Wolff

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IN MATHEMATICS the figure here depicted (a rectangle) is sort Figure l-l But the matter is not to be dismissed as simply as all that. ” And this is the first-though by no means the only or the most important-task of geometry. There are many other areas where familiarity with geometrical subject matter may interfere with our ability-at least initially-to think scientifically about geometry. Ask a layman to look at Figure l-2. It is drawn so that the two sides of the A Figure l-2 triangle issuing from the peak are equal.

A Figure l-5 But again what seems so obvious is not necessarily so. To show this, look at Figure l-5a and b. If A = B, then A and B are both right angles. If C = D, then C and D are both right angles also. But is it clear that A = C, or B = D? Nat necessarily. In the diagrams as drawn, we have in fact tried to make A not equal to C, and B not equal to D. The reader may object that the diagrams also look as though A and B were not equal, and as though C and D were not really equal. This is granted as far as looks are concerned; but geometry does not go by looks.

Furthermore, we may say or assume that the other words in the sentence, such as “that” and “has” are even less in need of a definition, because their meaning is self-evident and clear. It would be hard to maintain that the meaning of “part” is not well known. However, is it better known than the meaning of the term “point”? Why, in other words, define “point” in terms of other words which are claimed to be well known and unambiguous? Why not just claim that the meaning of the word “point” is well known and unambiguous and be done with it?