Über die Bedeutung des Weberschen Gesetzes. Beiträge zur Psychologie des Vergleichens und Messens. Von A. Meinong. Hamburg und Leipzig: Leopold Voss, 1896. Pp. 164

THE present work consists essentially of a single thesis proved by a single argument. The thesis is at once simple and ingenious, the argument at once lucid and subtle. The author avoids almost all the mistakes and confusions which beset writers on psychical measurement, and makes several important distinctions which are rarely, if ever, to be met with elsewhere.

Herr Meinong’s thesis is, briefly, as follows: The true import of Weber’s Law is, that equal dissimilarities (Verschiedenheiten) in the stimuli correspond to equal dissimilarities in the corresponding sensations; while the dissimilarity of two measurable quantities of the same kind may be regarded as measured by the difference of the logarithms of these quantities. Thus where sensations are what the author calls extensive, they are directly proportional to their stimuli, though wherever the sensations are quantitative, their dissimilarity is proportional, as in Fechner’s formula, to the difference of the logarithms of the stimuli – provided these be measurable quantities of the same kind. This double contention depends upon the distinction between dissimilarity (Verschiedenheit) and mathematical difference (Unterschied). The use made of this distinction demands a careful account of quantity and measurement, of indivisible quantities, and of relations which are quantities. I am unacquainted with any better discussion of these topics than that contained in the present volume, and the points where the author appears mistaken do not, I think, invalidate the most important part of his thesis.

The first section of the book consists of a discussion of the nature and range of quantity. It is pointed out that quantities need not be divisible, since relations may be quantities. Distance in space, for example, is unquestionably both a quantity and a relation: to suppose distance divisible, can only arise from a confusion between distance and length (Strecke). In like manner, the author continues, similarity and dissimilarity are quantities: two things may be more or less similar, but the similarity is certainly indivisible.

In this sweeping assertion that similarity and dissimilarity are always quantities, the author ignores an important controversy. Had he applied his doctrine to the relations of other pairs of terms than quantities of the same kind, it would, I think, have led him into serious errors. If the relations in question are reducible to identity and diversity of content, they cease to be properly quantities. Moreover this reduction is certainly valid in some cases. Herr Meinong asserts, for example, that between a colour and a tone there is more difference than between two colours (p. 44). It would be truer to say that there are more differences. Wherever the relations in question are reducible to complete identity in some points, and complete diversity in others, there quantity seems not properly applicable. Diversity of content appears to be incapable of quantity: we cannot say that diversity in respect of one content is equal or unequal to diversity in respect of another. But there are other cases – and it is to these, fortunately, that the author applies his doctrine – where a difference exists which is not reducible to mere diversity of content. Such cases are, among others, differences of position and of magnitude; and differences of magnitude, naturally, have the chief importance in discussing Weber’s Law.

The second section deals with comparison, especially as to magnitude. Apart from the possible objection that magnitude is a notion essentially dependent upon comparison, and that the present section ought, therefore, to have been the first, the account of quantitative comparison is excellent. Likeness and unlikeness are notions not demanding a definition; but they are not the only results of quantitative comparison, which is unique in yielding, not mere difference of magnitude, but the relations of greater and less. Whatever appears different, on immediate comparison, is different; but what is different only appears so down to a certain limit. Below this limit, a difference is imperceptible. Differences should not be described by their perceptibility, where such a description can be avoided; for our knowledge of the difference perceived is prior to our knowledge of the perception of difference, and a direct treatment of differences, where possible, is preferable to the indirect treatment by means of their perceptibility. Two just perceptible differences need not be equal; but we have a well-grounded presumption, in favour of their equality, where there is equal susceptibility to differences.

Comparison of parts and measurement form the subject of the third section. The author recognises, what is so often overlooked, that numerical measurement proper depends upon divisibility, and is therefore inapplicable to quantities which are relations. He points out, nevertheless, that, where indivisible quantities have divisible correlates, all the practical advantages of measurement may often be obtained by means of these correlates. Measurement proper is either mediate or immediate: the latter is only applicable to space and time. But there is, for intensive quantities, a third kind of measurement, which the author calls substitutive (surrogativ), because what is really measured is an extensive substitute. For example, distance, being a relation, is indivisible; but it is always associated with a length, which is divisible. Distances, therefore, are regarded as measured by means of the correlated lengths. Similarly velocities are regarded as measured by means of the lengths traversed in a given time. In such cases, though another quantity is really measured in place of the quantity in question, we regard the latter as measured, because the operation ensures one or more of the three advantages derived from measurement proper. These advantages are: (1) That an element of a continuum is replaced by a discrete term, namely a number, and the intractability of the continuum is relegated to the unit; (2) that the number thus obtained has the same relation of magnitude to other numbers as the correlated quantities have; (3) that the absolute limits, zero and infinity, which have validity for indivisible as well as for divisible quantities, are the same for the numbers and for the corresponding quantities. All these advantages are secured in measuring distances and velocities; the first only is secured in measuring temperature by the thermometer. This last case illustrates that measurement is not sharply separated from mere determination without measurement.

This excellent discussion of the sense in which indivisible quantities can be measured is applied, in the fourth section, to the measurement of the dissimilarity between quantities of the same kind. Dissimilarity is a relation, and therefore indivisible. In the case of two quantities of the same kind, their dissimilarity appears to be also a quantity.¹ With space and time, the distance is associated with an intervening length; but in some cases where dissimilarity is a quantity there is, according to Herr Meinong, no intervening length. By an intervening length he means, apparently, no more than the power of continuous variation from the one term to the other. As an instance where this is not possible, he gives the dissimilarity of a colour and a tone. This, however, is not properly a quantity, but a difference of content. In all cases where dissimilarity is a quantity, there must be, I think, an intervening length in the author’s sense. As, however, the subsequent discussion is confined to the dissimilarity of measurable quantities of the same kind, the above limitation does not impair the validity of the argument.

The dissimilarity of two quantities is evidently capable, at most, of a substitutive measurement. Where the quantities themselves are measurable, the dissimilarity must be measured, if measurable at all, by some function of the two quantities. This function is not the mathematical difference, for the dissimilarity is infinite when one of the quantities is zero and the other finite. Moreover, the mathematical difference is a radically distinct idea, dependent wholly on divisibility. Thus the difference of two lengths is a length, but their dissimilarity is a relation. The dissimilarity between 1 and 2 is greater than that between 6 and 7, though the difference is the same. Also the mathematical differences may differ when the dissimilarities are the same.

This distinction is certainly of great importance. It is one, moreover, which mathematics and preoccupation with spatiotemporal quantities tend to obscure. In finding a function for measuring dissimilarity, certain requirements are laid down. (1) The dissimilarity must vanish when the quantities are equal; (2) It must be infinite when one quantity is finite and the other is zero or infinite; (3) The dissimilarity between A and B plus that between B and C must be equal to that between A and C. These conditions are essentially similar to those which, in non-Euclidean Geometry, regulate the expression of distance in terms of co-ordinates, and Herr Meinong might have simplified a needlessly complicated piece of mathematics by reference to this analogous case. The conclusion is, that the function required is the logarithm of the ratio, just as, in non-Euclidean Geometry, it is the logarithm of the Anharmonic Ratio.² To this conclusion, if we remember the meaning of substitutive measurement, there seems no valid objection. It must be remembered that, in such measurement, the end to be attained is mainly practical – theoretically, the quantities in question are not measured at all. But there is a proposition, essential to Herr Meinong’s formula, which has great theoretical importance. If the dissimilarity of A and B is equal to that of C and D, then A, B, C, D are proportionals – a theorem which, if correct, throws a new light on Weber’s Law.

The fifth section deals with psychical measurement and the interpretation of Weber’s Law. This section is somewhat marred, I think, by a division of psychical quantities into extensive and intensive. The author does not accept the view that psychical quantities must be intensive. He urges, in agreement with Mr. Bradley,³ that psychical quantities may be extensive, since the presented, as such, is psychical. He does not explicitly proceed, like Mr. Bradley in a subsequent article, to infer that psychical states may be extended, but this inference seems irresistible. If the presented, as such, is psychical, then every possible object of experience is psychical. This leads either to the philosophy of Berkeley, or to an unknowable thing in itself. To urge, as Herr Meinong does, that imagined space is measurable and divisible, though purely psychical, seems either irrelevant or untrue. For imagined space is as little mental as real space; it differs from real space only in the fact that it does not exist: while the imagination of space, which does exist, is not divisible. We have an imagination of something divisible, but the imagination, which alone is mental, is not divisible. Such an argument, therefore, cannot prove the existence of psychical quantities which are divisible. This question is too wide for a review, but I cannot avoid the conviction that to regard the presented as necessarily psychical must make havoc of a most fundamental distinction.

The discussion of psychical measurement treats extensive and intensive psychical quantities separately. In the supposed case of the former, the author arrives at the conclusion that they are simply proportional to their stimuli. But his chief concern is with dissimilarities. Weber’s Law is regarded as showing that, if r₁ r₂ r₃ r₄ be four stimuli, and e_l e₂ e₃ e₄ the corresponding sensations, then if r₁: r₂ = r₃: r₄, the dissimilarity of e_l and e₂ is equal to that of e₃ and e₄. It follows from the previous section that equal dissimilarities of sensation correspond to equal dissimilarities of stimulus. No inference is possible, in general, as to the magnitude of the (intensive) sensations themselves: Fechner’s deduction of the logarithmic formula depends upon a confusion of difference and dissimilarity. The same confusion underlies the hypothesis, propped up by a so-called “law of relativity,” that the relative difference of two sensations is to be substituted for the absolute difference in Fechner’s deduction.

The discussion ends with a criticism of J. Merkel’s articles on the relation between stimulus and sensation.⁴ Merkel professes to prove experimentally that the sensation midway between two given qualitatively similar sensations corresponds to the arithmetic mean of the stimuli corresponding to the given sensations. On Herr Meinong’s hypothesis, it should correspond to the geometric mean, and he candidly confesses that his theory is incompatible with this result. But he is amply justified, I think, in holding such experiments to be inconclusive. Merkel supposed numerical measurement directly applicable to quantitative sensations, and accordingly regarded the idea of a mean sensation as perfectly definite. It must rather be held that such a discussion as Herr Meinong’s is necessary before such a phrase acquires any meaning. Merkel confesses (Phil. Stud. 10, p. 220) that a comparison of feelings of dissimilarity as such was not attempted, yet such a comparison, difficult as it would be in practice, is alone relevant to our author’s theory.

Many other points, which call for discussion, have been unnoticed, as not bearing directly on the argument of the work. There are throughout many subtle and suggestive observations on quantity, measurement, and relations. The author’s contention, if it perhaps simplifies the question, especially in the interpretation of Weber’s Law, with somewhat excessive optimism, is based on very close and careful reasoning, and offers, on essentials, very few vulnerable points.

B. Russell

*  Bertrand Russell, Review of Alexius Meinong, Über die Bedeutung des Weberschen Gesetzes, Mind, n.s. 8, no. 30 (Apr 1899), 251-6

¹ “Dissimilarity” is not quite an adequate translation of Verschiedenheit, but the word “difference” is required in the mathematical sense, and it is necessary to preserve the distinction by using different words for the two ideas.

²Cf., e.g., Whitehead, Universal Algebra, bk. 6, ch. 1.

³ “What Do We Mean by the Intensity of Psychical States?” Mind, 1895.

⁴ Phil. Stud. , 4, 5, 10.