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References
Spaier,La Pensée et la Qualité, p. 34.
It need be mentioned only in passing that mathematics cannot be regarded as exclusively the science of quantity. Its essence is the study of types of order, of which the quantitative one is a single instance.
“Die Axiome der Quantität”,Ber. d. Sächs. Gesellsch. d. Wiss., math.-phys. Klasse, 1901.
On the importance of the Archimedean axiom, see Hilbert, “Axiomatisches Denken”,Math. Annalen, Vol. 78, p. 408. The fact that one can estimate stellar distances by adjoining earthly ones, is not a logical consequence of theorems on congruence, but is an independent experimental conclusion.
So Warrain, a follower of Wronski, inQuantite, Infini, Continu, p. 9.
“Über die Bedeutung d. Weberschen Ges.” inGesam. Abhandl., Vol. 2, p. 219.
Cf. Duhem,La Theorie Physique, p. 219.
Dingler, inDas Experiment, p. 51 ff., has an important discussion in this connection, whatever one may think of his characterization of theoretical physics as without a physical meaning.
Runge, “Maß u. Messen” inEnzyk. d. math. Wiss., Bd. V, p. 4.
This definition of equality in terms of the relation generating the series is due to Campbell,Measurement and Calculation, p. 5. The many debts of this paper to Campbell are evident to all his readers, and cannot be made explicit even by repeated citations. The discussion in Duhem,Theorie Physique, p. 159 ff., takes similar lines, while Helmholtz's important essay onZaehlen u. Maessen, contains the germ of almost all subsequent works on the subject. The essay is reprinted in hisSchriflen zur Erkenntnistheorie, edited by Hertz and Schlick. “Equality between the comparable properties of two objects is an exceptional occurence, and can be recognized in empirical observations only by this, that the two equal objects make possible the noting under proper conditions of a special effect, not usually present in the interaction of other pairs of similar objects”; p. 85. See also Cournot,Essai..., p. 286 ff.
Cf. Runge,op. cit., p. 5 ff.
Technique of Controversy, pp. 162–74.
For the following illustration, see Campbell and Dudding, “Measurement of Light”,Philos. Magazine, 6 Ser. Vol. 44. The discussion in Lambert'sPhotometrie (translated in Ostwald's Klassiker) is the foundation for the whole science. He shows, incidentally, the self-corrective nature of scientific procedure. “If one wanted to investigate the validity of eye judgments, one must remember these psychological illusions in order to take cognizance of the remaining principles of photometry. But it is these principles that are presupposed in any investigation of the errors of eye judgments. Hence I do not see how a logical circle can be avoided if a rigorously proved photometry is desired. But if this rigor is abated a little, one can obtain the propositions of photometry with some degree of certainty.” Vol. 1. p. 7.
Walsh,Photometry, p. 7. Campbell points out that heat is capable of addition in a very restricted sense only, because no definition of addition will make the commutative and distributive axioms verifiable.Physics., p. 287.
Helmholtz,Op. cit., p. 89 ff.
Principles of Mathematics, p. 164 ff.
Op. cit., p. 166.
Couturat declares that sound common sense sides for the absolute theory. “It is reasonable to suppose that equal magnitudes have something more in common that those magnitudes of the same kind which are not equal; and it is contradictory to suppose that two equal magnitudes are differentiated from two unequal ones only by the fact of the unessential relation which connects them: equality in one case, inequality in the others.”Rev. de Met. et de Morale, 1904, p. 677. The reified concept is very evident in the “something in common”. Moreover, on Russell's theory there is a magnitude for every specific kind: a specific magnitude of pleasure for each grade of pleasure, a specific magnitude of density for each grade of density. Is the magnitude of density of two substances, whose densities are the same but are otherwise very dissimilark,two magnitudes orone? On the absolutee theory, since two magnitudes are of the same kind if one can be greater or less than the other, the answer is: one magnitude. But what determines whether two magnitudescan be compared? Is not the decision made, not by appealing to the magnitudes, but to the physical operations on quantities?
Op. cit., p. 178; cf. alsoAnalysis of Matter, p. 116.
Spaier,op. cit., pp. 0242–55.
Duhem,L'Evolution de la Mécanique, p. 199 ff.
“The ascertainment of qualitative features and relations is called measuring. We measure e. g. the length of different chords that have been put into a state of vibration, with an eye to the qualitative difference of the tones caused by their vibration, corresponding to this difference of length.” Hegel'sLogic (Wallace tr.), p. 200.
Erkenntnis II
Cf. Handbuch d. Physik, Bd. II, p. 5.
Buchanan,Poetry and Mathematics, p. 90.
Meinong,op. cit., p. 228, 275. The expression “surrogative measurement” is, of course, due to Meinong.
But one need not therefore confuse, as does Hegel, the distinction between temperature as intensive and the correlated expansion, which is extensive. Hegel,op. cit., p. 194.
Runge,op. cit., p. 8.
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This essay is the second chapter of a larger work on the logic of measurement
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Nagel, E., Hempel, C.G. Measurement. Erkenntnis 2, 313–335 (1931). https://doi.org/10.1007/BF02028166
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DOI: https://doi.org/10.1007/BF02028166