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With unusual depth and clarity, it covers the problem of the foundations of geometry, the theory of time, the theory and consequences of Einstein's relativity including: relations between theory and observations, coordinate definitions, relations between topological and metrical properties of space, the psychological problem of the possibility of a visual intuition of non-Euclidean structures, and many other important topics in modern science and philosophy.
While some of the book utilizes mathematics of a somewhat advanced nature, the exposition is so careful and complete that most people familiar with the philosophy of science or some intermediate mathematics will understand the majority of the ideas and problems discussed.
Partial CONTENTS: I. The Problem of Physical Geometry. Universal and Differential Forces. Visualization of Geometries. Spaces with non-Euclidean Topological Properties. Geometry as a Theory of Relations. II. The Difference between Space and Time. Simultaneity. Time Order. Unreal Sequences. Ill. The Problem of a Combined Theory of Space and Time. Construction of the Space-Time Metric. Lorentz and Einstein Contractions. Addition Theorem of Velocities. Principle of Equivalence. Einstein's Concept of the Problems of Rotation and Gravitation. Gravitation and Geometry. Riemannian Spaces. The Singular Nature of Time. Spatial Dimensions. Reality of Space and Time.
§ 1. THE AXIOM OF THE PARALLELS AND NON-EUCLIDEAN GEOMETRY
In Euclid's work, the geometrical achievements of the ancients reached their final form: geometry was established as a closed and complete system. The basis of the system was given by the geometrical axioms, from which all theorems were derived. The great practical significance of this construction consisted in the fact that it endowed geometry with a certainty never previously attained by any other science. The small number of axioms forming the foundation of the system were so self-evident that their truth was accepted without reservation. The entire construction of geometry was carried through by a skillful combination of the axioms alone, without any addition of further assumptions; the reliability of the logical inferences used in the proofs was so great that the derived theorems, which were sometimes quite involved, could be regarded as certain as the axioms. Geometry thus became the prototype of a demonstrable science, the first instance of a scientific rigor which, since that time, has been the ideal of every science. In particular, the philosophers of all ages have regarded it as their highest aim to prove their conclusions " by the geometrical method."
Euclid's axiomatic construction was also important in another respect. The problem of demonstrability of a science was solved by Euclid in so far as he had reduced the science to a system of axioms. But now arose the epistemological question how to justify the truth of those first assumptions. If the certainty of the axioms was transferred to the derived theorems by means of the system of logical concatenations, the problem of the truth of this involved construction was transferred, conversely, to the axioms. It is precisely the assertion of the truth of the axioms which epitomizes the problem of scientific knowledge, once the connection between axioms and theorems has been carried through. In other words: the implicational character of mathematical demonstrability was recognized, i.e., the undeniable fact that only the implication "if a, then b" is accessible to logical proof. The problem of the categorical assertion "a is true b is true", which is no longer tied to the "if", calls for an independent solution. The truth of the axioms, in fact, represents the intrinsic problem of every science. The axiomatic method has not been able to establish knowledge with absolute certainty; it could only reduce the question of such knowledge to a precise thesis and thus present it for philosophical discussion.
This effect of the axiomatic construction, however, was not recognized until long after Euclid's time. Precise epistemological formulations could not be expected from a naive epoch, in which philosophy was not yet based upon well-developed special sciences, and thinkers concerned themselves with cruder things than the truth of simple and apparently self-evident axioms. Unless one was a skeptic, one was content with the fact that certain assumptions had to be believed axiomatically; analytical philosophy has learned mainly through Kant's critical philosophy to discover genuine problems in questions previously utilized only by skeptics in order to deny the possibility of knowledge. These questions became the central problems of epistemology. For two thousand years the criticism of the axiomatic construction has remained within the frame of mathematical questions, the elaboration of which, however, led to peculiar discoveries, and eventually called for a return to philosophical investigations.
The mathematical question concerned the reducibility of the axiomatic system, i.e., the problem whether Euclid's axioms represented ultimate propositions or whether there was a possibility of reducing them to still simpler and more self-evident statements. Since the individual axioms were quite different in character with respect to their immediacy, the question arose whether some of the more complicated axioms might be conceived as consequences of the simpler ones, i.e., whether they could be included among the theorems. In particular, the demonstrability of the axiom of the parallels was investigated. This axiom states that through a given point there is one and only one parallel to a given straight line (which does not go through the given point), i.e., one straight line which lies in the same plane with the first one and does not intersect it. At first glance this axiom appears to be self-evident. There is, however, something unsatisfactory about it, because it contains a statement about infinity; the assertion that the two lines do not intersect within a finite distance transcends all possible experience. The demonstrability of this axiom would have enhanced the certainty of geometry to a great extent, and the history of mathematics tells us that excellent mathematicians from Proclus to Gauss have tried in vain to solve the problem.
A new turn was given to the question through the discovery that it was possible to do without the axiom of parallels altogether. Instead of proving its truth the opposite method was employed: it was demonstrated that this axiom could be dispensed with. Although the existence of several parallels to a given line through one point contradicts the human power of visualization, this assumption could be introduced as an axiom, and a consistent geometry could be developed in combination with Euclid's other axioms. This discovery was made almost simultaneously in the twenties of the last century by the Hungarian, Bolyai, and the Russian, Lobatschewsky; Gauss is said to have conceived the idea somewhat earlier without publishing it.
But what can we make of a geometry that assumes the opposite of the axiom of the parallels? In order to understand the possibility of a non-Euclidean geometry, it must be remembered that the axiomatic construction furnishes the proof of a statement in terms of logical derivations from the axioms alone. The drawing of a figure is only a means to assist visualization, but is never used as a factor in the proof; we know that a proof is also possible by the help of "badly-drawn" figures in which so-called congruent triangles have sides obviously different in length. It is not the immediate picture of the figure, but a concatenation of logical relations that compels us to accept the proof. This consideration holds equally well for non-Euclidean geometry; although the drawing looks like a " badly- drawn " figure, we can with its help discover whether the logical requirements have been satisfied, just as we can do in Euclidean geometry. This is why non-Euclidean geometry has been developed from its inception in an axiomatic construction; in contradistinction to Euclidean geometry where the theorems were known first and the axiomatic foundation was developed later, the axiomatic construction was the instrument of discovery in non- Euclidean geometry.
With this consideration, which was meant only to make non-Euclidean geometry plausible, we touch upon the problem of the visualization of geometry. Since this question will be treated at greater length in a later section, the remark about "badly-drawn" figures should be taken as a passing comment. What was intended was to stress the fact that the essence of a geometrical proof is contained in the logic of its derivations, not in the proportions of the figures. Non-Euclidean geometry is a logically constructible system—this was the first and most important result established by its inventors.
It is true that a strict proof was still missing. No contradictions were encountered—yet did this mean that none would be encountered in the future? This question constitutes the fundamental problem concerning an axiomatically constructed logical system. It is to be expected that non-Euclidean statements directly contradict those of Euclidean geometry; one must not be surprised if, for instance, the sum of the angles of a triangle is found to be smaller than two right angles. This contradiction follows necessarily from the reformulation of the axiom of the parallels. What is to be required is that the new geometrical system be self-consistent. The possibility can be imagined that a statement a, proved within the non- Euclidean axiomatic system, is not tenable in a later development, i.e., that the statement not-a as well as the statement a is provable in the axiomatic system. It was incumbent upon the early adherents of non-Euclidean geometry, therefore, to prove that such a contradiction could never happen.
The proof was furnished to a certain extent by Klein's Euclidean model of non-Euclidean geometry. Klein succeeded in coordinating the concepts of Euclidean geometry, its points, straight lines, and planes, its concept of congruence, etc., to the corresponding concepts of non-Euclidean geometry, so that every statement of one geometry corresponds to a statement of the other. If in non-Euclidean geometry a statement a and also a statement not-a could be proved, the same would hold for the coordinated statements a' and not-a' of Euclidean geometry; a contradiction in non-Euclidean geometry would entail a corresponding contradiction in Euclidean geometry. The result was a proof of consistency, the first in the history of mathematics : it proceeds by reducing a new system of statements to an earlier one, the consistency of which is regarded as virtually certain.
After these investigations by Klein the mathematical significance of non-Euclidean geometry was recognized. Compared with the natural geometry of Euclid, that of Bolyai and Lobatschewsky appeared strange and artificial; but its mathematical legitimacy was beyond question. It turned out later that another kind of non-Euclidean geometry was possible. The axiom of the parallels in Euclidean geometry asserts that to a given straight line through a given point there exists exactly one parallel; apart from the device used by Bolyai and Lobatschewsky to deny this axiom by assuming the existence of several parallels, there was a third possibility, that of denying the existence of any parallel. However, in order to carry through this assumption consistently, a certain change in a number of Euclid's other axioms referring to the infinity of a straight line was required. By the help of these changes it became possible to carry through this new type of non- Euclidean geometry.
As a result of these developments there exists not one geometry but a plurality of geometries. With this mathematical discovery, the epistemological problem of the axioms was given a new solution. If mathematics is not required to use certain systems of axioms, but is in a position to employ the axiom not-a as well as the axiom a, then the assertion a does not belong in mathematics, and mathematics is solely the science of implication, i.e., of relations of the form "if ... then"; consequently, for geometry as a mathematical science, there is no problem concerning the truth of the axioms. This apparently unsolvable problem turns out to be a pseudo-problem. The axioms are not true or false, but arbitrary statements. It was soon discovered that the other axioms could be treated in the same way as the axiom of the parallels. "Non-Archimedian," "non-Pascalian," etc., geometries were constructed; a more detailed exposition will be found in § 14.
These considerations leave us with the problem into which discipline the question of the truth of the assertion a should be incorporated. Nobody can deny that we regard this statement as meaningful; common sense is convinced that real space, the space in which we live and move around, corresponds to the axioms of Euclid and that with respect to this space a is true, while not-a is false. The discussion of this statement leads away from mathematics; as a question about a property of the physical world, it is a physical question, not a mathematical one. This distinction, which grew out of the discovery of non-Euclidean geometry, has a fundamental significance: it divides the problem of space into two parts; the problem of mathematical space is recognized as different from the problem of physical space.
It will be readily understood that the philosophical insight into the twofold nature of space became possible only after mathematics had made the step from Euclid's geometry to non-Euclidean geometries. Up to that time physics had assumed the axioms of geometry as the self-evident basis of its description of nature. If several kinds of geometries were regarded as mathematically equivalent, the question arose which of these geometries was applicable to physical reality; there is no necessity to single out Euclidean geometry for this purpose. Mathematics shows a variety of possible forms of relations among which physics selects the real one by means of observations and experiments. Mathematics, for instance, teaches how the planets would move if the force of attraction of the sun should decrease with the second or third or nth power of the distance; physics decides that the second power holds in the real world. With respect to geometry there had been a difference; only one kind of geometry had been developed and the problem of choice among geometries had not existed. After the discoveries of non-Euclidean geometries the duality of physical and possible space was recognized. Mathematics reveals the possible spaces; physics decides which among them corresponds to physical space. In contrast to all earlier conceptions, in particular to the philosophy of Kant, it becomes now a task of physics to determine the geometry of physical space, just as physics determines the shape of the earth or the motions of the planets, by means of observations and experiments.
But what methods should physics employ in order to come to a decision? The answer to this question will at the same time supply an answer to the question why we are justified in speaking of a specific physical space. Before this problem can be investigated more closely, another aspect of geometry will have to be discussed. For physics the analytic treatment of geometry became even more fruitful than the axiomatic one.
§ 2. RIEMANNIAN GEOMETRY
Riemann's extension of the concept of space did not start from the axiom of the parallels, but centered around the concept of metric.
Riemann developed further a discovery by Gauss according to which the shape of a curved surface can be characterized by the geometry within the surface. Let us illustrate Gauss' idea as follows. We usually characterize the curvature of the surface of a sphere by its deviation from the plane; if we hold a plane against the sphere it touches only at one point; at all other points the distances between plane and sphere become larger and larger. This description characterizes the curvature of the surface of the sphere "from the outside"; the distances between the plane and the surface of the sphere lie outside the surface and the decision about the curvature has to make use of the third dimension, which alone establishes the difference between curved and straight. Is it possible to determine the curvature of the surface of the sphere without taking outside measurements? Is it meaningful to distinguish the curved surface from the plane within two dimensions? Gauss showed that such a distinction is indeed possible. If we were to pursue "practical geometry" on the sphere, by surveying, for instance, with small measuring rods, we should find out very soon that we were living on a curved surface. For the ratio of circumference u and diameter d of a circle we would obtain a number smaller than π = 3.14 ... as is shown in Fig. 1. Since we stay on the surface all the time, we would not measure the "real diameter" which cuts through the inner part of the sphere, but the "curved diameter" which lies on the surface of the sphere and is longer. This diameter divided into the circumference results in a number smaller than π. Nevertheless, it is meaningful to call the point M "the center of the circle on the surface of the sphere" because it has the same distance from every point of the circle; that we find ourselves on a sphere is noticed by means of the deviation of the ratio from π. In this way we obtain a geometry of a spherical surface which is distinguished from the ordinary geometry by the fact that different metrical relations hold for this kind of geometry. In addition to the change in the ratio between circumference and diameter of a circle, an especially important feature is that the sum of the angles of a triangle on a sphere is greater than 180°.
Excerpted from THE PHILOSOPHY OF SPACE & TIME by HANS REICHENBACH, Maria Reichenbach, JOHN FREUND. Copyright © 1958 Maria Reichenbach. Excerpted by permission of Dover Publications, Inc..
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