Gaps in Euclid’s Arguments 13 situation. The light coming from a certain point A in the middle of the loop-shaped galaxy follows two paths to our eyes and along the way makes two different dots (B and C ) on the photographic plate. As a result, the three points A, B, and C form a triangle whose interior angle measures add up to a number slightly greater than 180ı. (Although it doesn’t look like a triangle in this diagram, remember that the edges are paths of light rays. What could be straighter than that?) Yet Euclidean geometry predicts that every triangle has interior angle measures that add up to exactly 180ı. We can see why Euclid’s arguments fail in this situation by examining the figure: in this case there are two distinct line segments connecting the point A to the observer’s eye, which contradicts Euclid’s intended meaning of his first postulate. We have no choice but to conclude that the geometry of the physical world we live in does not exactly follow Euclid’s rules. A B C Fig. 1.5. A triangle whose angle sum is greater than 180ı. Gaps in Euclid’s Arguments As a result of the non-Euclidean breakthroughs of Lobachevsky, Bolyai, and Gauss, math- ematicians were forced to undertake a far-reaching reexamination of the very foundations of their subject. Euclid and everyone who followed him had regarded postulates as self- evident truths about the real world, from which reliable conclusions could be drawn. But once it was discovered that two or three conflicting systems of postulates worked equally well as logical foundations for geometry, mathematicians had to face an uncomfortable question: what exactly are we doing when we accept some postulates and use them to prove theorems? It became clear that the system of postulates one uses is in some sense an arbitrary choice once the postulates have been chosen, as long as they don’t lead to any contradictions, one can proceed to prove whatever theorems follow from them. Thus was born the notion of a mathematical theory as an axiomatic system—a se- quence of theorems based on a particular set of assumptions called postulates or axioms (these two words are used synonymously in modern mathematics). We will give a more precise definition of axiomatic systems in the next chapter. Of course, the axioms we choose are not completely arbitrary, because the only ax- iomatic systems that are worth studying are those that describe something useful or inter- esting—an aspect of the physical world, or a class of mathematical structures that have proved useful in other contexts, for example. But from a strictly logical point of view, we may adopt any consistent system of axioms that we like, and the resulting theorems will constitute a valid mathematical theory.

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