What May We Take for Granted?

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. What May We Take for Granted?

What May We Take for Granted?

This is all very fine; but how does the geometer know what postulates to lay down? One is tempted to say that he is at liberty to postulate anything he pleases, and investigate the results; and that whether or not his postulate ever be realized, the propositions that he deduces from it, being true, are of scientific interest. Actually, however, it is not quite as simple as all that. If it were sufficient to make a single postulate it would be as simple as all that; but it turns out that this is not sufficient any more than it is sufficient to have a single undefined term. We must have several postulates; and they must be such, as a whole, that a geometry flows out of them. The requirements are three.

In the first place, the system of postulates must be “categorical” or complete—there must be enough of them, and they must cover enough ground, for the support of a complete system of geometry. In practice the test for this is direct. If we got to a point in the building up of a geometry where we could not prove whether a certain thing was one way always, or always the other way, or sometimes one way and sometimes the other, we should conclude that we needed an additional postulate covering this ground directly or indirectly. And we should make that postulate—because it is precisely the [123]things that we can’t prove which, in practical work, we agree to assume. Even Euclid had to adopt this philosophy.

In the second place, the system of postulates must be consistent—no one or more of them may lead, individually or collectively, to consequences that contradict the results or any other or others. If in the course of building up a geometry we find we have proved two propositions that deny one another, we search out the implied contradiction in our postulates and remedy it.

Finally, the postulates ought to be independent. It should not be possible to prove any one of them as a consequence of the others. If this property fails, the geometry does not fail with it; but it is seriously disfigured by the superfluity of assumptions, and one of them should be eliminated. If we are to assume anything unnecessarily, we may as well assume the whole geometry and be done with it.

The geometer’s business then is to draw up a set of postulates. This he may do on any basis whatever. They may be suggested to him by the behavior of points, lines and planes, or by some other concrete phenomena; they may with equal propriety be the product of an inventive imagination. On proceeding to deduce their consequences, he will discover and remedy any lack of categoricity or consistence or independence which his original system of postulates may have lacked. In the end he will have so large a body of propositions without contradiction or failure that he will conclude the propriety of his postulates to have been established, and the geometry based on them to be a valid one.

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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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