ON THE THEORY OF RELATIVITY: ANALYSIS OF THE POSTULATESby R. D. Carmichael
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This analysis of the postulates of relativity was undertaken in order to ascertain on just which of the postulates certain fundamental conclusions of the
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This analysis of the postulates of relativity was undertaken in order to ascertain on just which of the postulates certain fundamental conclusions of the theory depend. A moment's reflection will convince one of the importance of such an analysis. Some of the conclusions of relativity have been attacked by those who admit just the parts of the postulates from which the conclusions objected to can be derived by purely logical processes. In this paper I have sought to establish some of the most fundamental and most readily accessible conclusions of the theory on the smallest possible foundation from the postulates. This plan of treatment, instead of giving rise to more complicated arguments than those hitherto usually employed, has had the opposite effect of leading to increased simplicity both in the notions which enter and in the arguments by which proofs are reached.
1. In the theory of relativity the word "postulate" has been used in the sense in which one is accustomed to employ the term "law of nature."
When the work was taken in hand it soon became evident that there was something to be done both on the postulates themselves and on the very first theorems which are to be deduced from them, as the reader will see by reference to the treatment below. It thus appears that some of the most striking conclusions of the theory depend on only a part of the postulates. To bring this fact prominently into view in one's mind is to put the whole subject in a clearer light where one may see better the interactions of its parts and its general relations to the whole body of scientific and philosophical knowledge.
A certain method has been consistently employed throughout the paper to indicate the postulates on which each theorem depends. Each postulate is designated by a letter. At the end of a theorem and enclosed in parentheses are references (by means of these letters) to the postulates on which the theorem as demonstrated depends. Thus theorem I. depends on postulates M and R'.
2. This method has been employed by Veblen and Young in their Projective Geometry, 1910.
In carrying out the initial purpose of the paper an important part of the general fundamentals of the theory of relativity come in for a fresh development along lines more or less new. It was observed that the addition to this essential matter of a relatively small amount of material would make the paper as a whole serve as an elementary introduction to the entire theory; and consequently such matter as was necessary to this end has been incorporated. I felt the more disposed to do this in view of the fact that even this additional material is presented in a somewhat novel manner. It is believed that the paper in its present form may be read profitably by one who is making his first acquaintance with the theory and that it will afford an easier introduction than any which has yet been offered.
In every body of doctrine which consists of a finite number of postulates and their logical consequences there are necessarily certain theorems which have the following fundamental relation to the whole body of doctrine: By means of one of these theorems and all the postulates but one that remaining postulate may be demonstrated. That is, one may assume such a theorem in place of one of the postulates and then demonstrate that postulate. When the postulate has thus been proved it may be used in argument as well as the theorem itself; hence it is clear that all the consequences which were obtained from the first set of postulates may now be deduced again, though perhaps in a somewhat different manner. That is, if we consider the whole body of doctrine, composed of postulates and theorems, this totality is the same in the two cases. Two sets of postulates which thus give rise to the same body of doctrine (consisting of postulates and theorems together) are said to be logically equivalent.
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- THE PHYSICAL REVIEW: A Journal Of Experimental And Theoretical Physics , #35
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