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The Philosophy of Space and Time (Dover Books on Physics) Paperback – 17 Mar 2003
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About the Author
"John E. Freund." Professor of Mathematics Emeritus Arizona State University Educated at the University of London, U.C.L.A., Columbia University, and the University of Pittsburgh, Doctor Freund's interest in Mathematics, Logic, and the Philosophy of Science led him to a career in statistics. Keynoted by his approach to statistics as a way of thinking, and as such a refinement of everyday thinking, his textbooks in statistics at various levels and for various fields of application have been bestsellers for fifty years.
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If you have been confused by discussions of rigid rods, clocks, simultaneity etc. in other sources, check out Reichenbach's construction of the light geometry and his discussion of the indefinite space type. Want to understand how gravity affects spacetime but do not want to study differential geometry? Read Reichenbach's sections on the Riemannian spaces and his chapter about space and time in gravitational fields. No other source explains these relationships as clearly and without resorting to silly or trivial examples.
A beautiful scholarly book which is thoroughly accessible. The author's great love of the subject is much in evidence.
Reichenbach, in the Introduction, rues the current estrangement of philosophy and science, longing for the "natural philosophy" of the past, where thinkers were well-versed in both areas.
So this book takes us through the philosophy of space and time accompanied and supported by empirical and theoretical scientific work. He seems to have little in the way of agenda or "-isms" to tout, nor is he inclined to spend much ink on rehashing historical debates or trivial examples. And although the book winds it way eventually to General Relativity, we are thankfully not dragged through the typical "Aristotlean view -> Galilean view -> Einsteinian view" that is so commonly used.
Instead, he begins by discussing Euclidean space, the nature of geometry and so on. Throughout, the notion of topology is a common thread. Time, simultaneity, Lorentz, Principle of Equivalence, and gravitational effects on the topology of spacetime, are some of the steps through the book.
In section 39, for example, he guides us on a detour entitled "The Analytic Treatment of Reimannian Spaces", just to carry "...the treatment of general geometry a little further." In four short pages and a modicum of equations, the nature of tensors as a natural mathematical consequence appear, effortlessly and painlessly.
All along, woven in, are cogent philosophical treatments of the topic currently under discussion. The book is a good example of the author's desire to see philosophy and science melded again, and good example of his prowess in both areas.
He wrote in the Introduction to this 1927 book, “The path of the present philosophical work led therefore through the natural sciences… It is hoped… to give an account of the superiority of a philosophical method closely connected with the results of empirical science… Modern scientific epistemology therefore justifies discoveries of such far-reaching consequences as would, in former times, have been merely empty speculation… This book has been written in the knowledge that solutions are attainable. It is intended… to present in a comprehensive fashion the treasure of philosophical results that has become the common property of scientific philosophy… and also to go beyond it on new paths that were opened … through a persistent analysis of mathematical physics… one may see as the noblest aim of scientific philosophy the establishment of the concept of objective truth as the ultimate criterion of all philosophical knowledge.”
He says in the first chapter, “Non-Euclidean geometry is a logically constructible system---this was the first and most important result established by its inventors.” (Pg. 4) He adds, “there exists no one geometry but a plurality of geometries. With this mathematical discovery, the epistemological problem of the axioms was given a new solution… 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… These considerations leave us with the problem into which discipline the questions of the truth of the assertion ‘a’ should be incorporated. Nobody can deny that… 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.” (Pg. 5-6)
He explains, “The images by which we visualize geometry are always so adjusted as to correspond to the laws which we read from them; these laws are always implied. That statement that we cannot visualize non-Euclidean geometry must therefore be reformulated. We cannot visualize non-Euclidean geometry by means of Euclidean elements of visualization. In this form the result is trivial; what it denies is a logical impossibility. The question must be asked differently: Can we change the image-producing elements in such a way that we can read the laws of non-Euclidean geometry from the new images? Only in such a manner can we attempt a visualization of non-Euclidean geometry.” (Pg. 43-44)
He argues, “Arguments that present Euclidean space as ‘reasonable’ or ‘given by nature’ must not be employed to establish a preference for a certain kind of mathematical space. They may be used in favor of the choice of Euclidean space for physics, in which case we might add that they also speak in favor of the opposite choice, since physical space is non-Euclidean according to Einstein. The visual preference for Euclidean space therefore cannot depend on its special suitability for the visualization of natural objects, but rather on an INHERENT PROPERTY that has no connection with the outside world.” (Pg. 83)
He says, “We can summarize our results as follows: There is no pure visualization in the sense of the a priori philosophies; every visualization is determined by previous space perceptions, and any separation into perceptual space and space of visualization is not permissible, since the specifically visual elements of the imagination are derived from perceptual space. What led to the mistaken conception of pure visualization was rather an improper interpretation of the normative function, which we have recognized… as an essential element in all visual representations. Indeed. All arguments which have been introduced for the distinction of perceptual space and space of visualization are based on just this normative component of the imagination.” (Pg. 90-91)
He points out, “Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physics, its effect in the area of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential go guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension, it does not lose in any way its peculiar character as time. Through the combination of space and time into a four-dimensional manifold we merely express the fact that it takes four numbers to determine a world event, namely three numbers for the spatial location and one for time.” (Pg. 110)
He observes, “Thus we are faced with a circular argument. To determine the simultaneity of distant events we need to know a velocity, and to measure a velocity we require knowledge of the simultaneity of distant events. The occurrence of this circularity proves that simultaneity is not a matter of knowledge, but a coordinative definition, since the logical circle shows that a knowledge of simultaneity is impossible in principle.” (Pg. 126-127)
He states, “We define: any two events which are indeterminate as to their time order may be called simultaneous… The concept ‘simultaneous’ is to be reduced to the concept ‘indeterminate as to time order.’ This result supports our intuitive understanding of the concept simultaneous. Two simultaneous events are so situated that a causal chain cannot travel from one to the other in either direction. Events which occur at this moment in a distant land can no longer be influenced by us… conversely, they can have no effect on what is happening here at the present moment. Simultaneity means the exclusion of causal connection.” (Pg. 145)
He explains, “The physical core of the [relativity] theory… consists of the hypothesis that natural measuring instruments follow coordinative definitions different from those assumed in the classical theory. This statement is, of course, empirical. On its truth depends only the physical theory of relativity. However, the philosophical theory of relativity. i.e., the discover of the additional character of the metric in all its details, holds independently of experience. Although it was developed in connection with physical experiments, it constitutes a philosophical result not subject to the criticism of the individual sciences.” (Pg. 177)
He summarizes, “Time, and through it causality, supplies the measure and the order of space; not time order alone, but the combined space-time order reveals itself as the ordering scheme governing causal chains and thus as the expression of the causal structure of the universe.” (Pg. 268)
This is a really EXCELLENT explanation of the principles of Relativity, and non-Euclidean geometry, that will be very helpful to anyone looking into these subjects.
I would recommend this book to those curious about the meaning of "space time" with the slight caveat that Reichenbach was a Positivist, and there are many who disagree with the basic philosophy of Positivism.
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