Yesterday, I posted a talk in which I invoked Euclid to show how all conclusions rest on assumptions. Soon after I finished that, I pulled a book off the shelf at home, one of those books I've been meaning to read for more years than I care to say. It opens by making the same point, with more elegance:
In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry and you remember—perhaps with more respect than love—the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard every one with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if someone were to ask you: “What, then, do you mean by the assertion that these propositions are true?“ Let us proceed to give this question a little consideration.
Geometry sets out from certain conceptions such as “plane,” “point,” and “straight line,” with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as “true.” Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct (“true”) when it has been derived in the recognised manner from the axioms. The question of the “truth” of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called “straight lines,” to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept “true” does not tally with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.
The book, as it was described to me by my 9th-grade science teacher, is Relativity, by A. Einstein.
(For Einstein, Euclid is no idle metaphor. Relativity is founded on the assumption that one of Euclid's axioms is false. Because gravity warps spacetime, seemingly parallel lines can diverge from one another.)