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. THAT PARALLEL POSTULATE

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Modern Geometric Methods; the Dividing Line Between Euclidean and Non-Euclidean; and the Significance of the Latter

BY THE EDITOR

The science of geometry has undergone a revolution of which the outsider is not informed, but which it is necessary to understand if we are to attain any comprehension of the geometric formulation of Einstein’s results; and especially if we are to appreciate why it is proper and desirable to formulate these results geometrically at all. The classical geometer regarded his science from a narrow viewpoint, as the study of a certain set of observed phenomena—those of the space about us, considered as an entity in itself and divorced from everything in it. It is clear that some things about that space are not as they appear (optical illusions), and that other things about it are true but by no means apparent (the sum-of-squares property of a right triangle, the formulæ for surface and volume of a sphere, etc.). While many things about space are “obvious,” these need in the one case disproof and in the other discovery and proof. With all their love of mental processes for their own sake, it is [112]then not surprising that the Greeks should have set themselves the task of proving by logical process the properties of space, which a less thoughtful folk would have regarded as a subject only for observational and experimental determination.

But, abstract or concrete, the logical structure must have a starting point; and it is fair to demand that this consist in a statement of the terms we are going to use and the meanings we are going to attach to them. In other words, the first thing on the program will be a definition, or more probably, several definitions.

Now the modern scientist has a somewhat iconoclastic viewpoint toward definitions, and especially toward the definition of his very most fundamental ideas.

We do not speak here in terms of dictionary definitions. These have for object the eminent necessity of explaining the meaning and use of a word to some one who has just met it for the first time. It is easy enough to do this, if the doer possesses a good command of the language. It is not even a matter of grave concern that the words used in the definition be themselves known to the reader; if they are not, he must make their acquaintance too. Dr. Johnson’s celebrated definition of a needle stands as perpetual evidence that when he cannot define a simple thing in terms of things still simpler, the lexicographer is forced to define it in terms of things more complex. Or we might demonstrate this by noting that the best dictionaries are driven to define such words as “and” and “but” by using such complex [113]notions as are embodied in “connective,” “continuative,” “adversative,” and “particle.”

It is otherwise with the scientist who undertakes to lay down a definition as the basis of further procedure in building up the tissue of his science. Here a degree of rigorous logic is called for which would be as superfluous in the dictionary as the effort there to attain it would be out of place. The scientist, in building up a logical structure that will withstand every assault, must define everything, not in terms of something which he is more or less warranted in supposing his audience to know about, but actually in terms of things that have already been defined. This really means that he must explain what he is talking about in terms of simpler ideas and simpler things, which is precisely what the lexicographer does not have to worry about. This is why it is quite trivial to quote a dictionary definition of time or space or matter or force or motion in settlement of a controversy of scientific or semi-scientific nature.

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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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