Abstract
Referring to my philosophical account of physical geometry and chronometry, H. Putnam1 says (p. 211): “Grünbaum has in my opinion failed to give a true picture of one of the greatest scientific advances of all time.” I have not found one single argument in Putnam’s 50 pages that could serve to sustain this judgment. Nay, reflection on fundamental mathematical and physical errors on which he rests much of his case has enabled me to uncover substantial new support for my position from the general theory of relativity and from elsewhere in physics.
I wish to thank my physicist colleague Allen I. Janis for invaluable help with points and examples in Sections 2, 8 and 9, and I am likewise indebted to Wesley C. Salmon for discussions relating to the interpretation of Putnam’s views. By prior arrangement, a slightly different version of this paper will appear as part of the author’s book Geometry and Chronometry in Philosophical Perspective, to be published in 1968 by the University of Minnesota Press, Minneapolis, Minn. I am most grateful to Hilary Putnam for having written the essay to which the present paper is a response; his work has been a valuable stimulus to me to clarify my views both to others and to myself. Critical severity is linked here with friendly respect.
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References
[37]. All my subsequent references to Putnam will be to this essay of his, and the pages from which my citations of it are drawn will be specified within ordinary parentheses in my text immediately preceding the quoted passages.
Putnam’s critique is addressed mainly to what I wrote in ‘Geometry, Chronometry, and Empiricism’, which first appeared in Minnesota Studies in the Philosophy of Science, vol. III: Scientific Explanation, Space, and Time (ed. by H. Feigl and G. Maxwell) Minneapolis 1962, pp. 405–526. Unless otherwise specified, all references to previously published statements of mine that are criticized by Putnam are to this 1962 essay. And in order to distinguish references to it from those to Putnam’s essay, the numeral ‘1962’ will precede the appropriate page number of my essay within parentheses. Since this 1962 essay will be republished as Chapter I of my 1968 book Geometry and Chronometry in Philosophical Perspective, the additional reference ‘1968, ch. I’ may also occur in the same parentheses.
Cf. [33], p. 238.
There are also non-linear transformations between these coordinates which leave the light-velocity invariant. Cf. [40], pp. 172-175.
For the general principles relevant to the rotating rod lengths in system I, cf. [10], § 3, p. 116, and [33], p. 223.
We shall have occasion to discuss t-time light velocities on the rotating disk more fully in Section 8.2.
Thus P. Bergmann writes ([3], p. 158): “We can formulate the special theory of relativity in terms of curvilinear coordinate systems and general coordinate transformations in a four dimensional world” and ([2], p. 207) “A non-inertial frame of reference in the special theory of relativity … will include both rectilinear and curvilinear coordinate systems engaged in arbitrary motion”. As A. Janis has noted, this introduction of non-inertial frames into the STR is entirely analogous to their introduction into Newtonian mechanics: when Newton’s second law of motion is referred to non-inertial frames, it ceases to have the form F = ma, and the statement of the law then includes Coriolis and centrifugal terms among others ([36], p. 104). — The Riemann curvature-tensor is constructed solely from the components of the 4-tensor gik and from their first and second derivatives with respect to the coordinates. And this tensor enables us to speak of the gravitational field in an absolute sense. Thus J. L. Synge writes ([45], p. IX): “In Einstein’s theory, either there is a gravitational field or there is none, according as the Riemann tensor does not or does vanish. This is an absolute property; it has nothing to do with any observer’s world-line. Space-time is either flat or curved, and [one must] … separate truly gravitational effects due to curvature of space-time from those due to curvature of the observer’s world-line (in most ordinary cases the latter predominate).”
Quoted in [48], p. 170.
Theories in which the mechanical mass of, say, an electron turns out to be entirely electromagnetic (cf. [36], p. 528) are, of course, excluded here as incompatible with Putnam’s purpose to use the term ‘mass’ in intensionally alternative ways.
In the text that I omitted between the two sentences cited here, Putnam claims to correct “a major error” in my writings on Zeno’s paradoxes. I shall deal with this allegation in Section 5 below.
I disregard as irrelevant for now the broader usage of ‘GC’ in which the Chronometric part of the thesis asserts the conventionality of metrical simultaneity as well as of temporal congruence, since simultaneity will be treated in Section 8 below.
I am indebted to my mathematical colleague Albert Wilansky for clarifying comments on the presumed meaning of some of Putnam’s remarks.
(1962, p. 413, fn. 5). The two references which I give in that footnote also make it quite clear that my statements in that footnote are predicated on the countable additivity of the standard mathematical theory.
The argument which is about to be given by reference to this approximate form of the law can be readily generalized to forms of the law which allow for the temperature-dependence of the rate of thermal expansion and hence involve more than one coefficient of thermal expansion.
“This depends upon the identity of gravitational and inertial mass. The point is that in order to make the interior gravitational forces approach zero, we must make the mass of B approach zero and hence F = ma→0.”
Editors’ Note
The quoting of extended passages from Hilary Putnam’s essay ‘An Examination of Grünbaum’s Philosophy of Geometry’ has been done with the kind permission of John Wiley and Sons. The passages from Albert Einstein’s essay ‘Geometry and Experience’, which first appeared in his Sidelights on Relativity (in the translation of G. B. Jeffery and W. Perrett), were quoted with the kind permission of E. P. Dutton and Co., New York, and Methuen and Co. Ltd., London.
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Grünbaum, A. (1969). Reply to Hilary Putnam’s ‘An Examination of Grünbaum’s Philosophy of Geometry’. In: Cohen, R.S., Wartofsky, M.W. (eds) Boston Studies in the Philosophy of Science. Boston Studies in the Philosophy of Science, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3381-7_1
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