Abstract
The shortest answer to the question about what kind of unification is present in the totality lies in a direct reference to the phenomena. And here we truly are concerned with ultimate facts. But we are not thereby relieved of the task of considering this kind of combination more carefully, in order to bring into relief its characteristic differences from other kinds, especially since false characterizations and confusions of it with other species of relations have been an all too common occurrence. To this end we shall test a series of possible theories, some of which have actually been advanced. Each of these theories characterizes the collective unification in a different way and, in relation thereto, seeks also to explain in a different way the origin of the concepts multiplicity and number.
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Notes
Perhaps some readers will be surprised at my indecisive language here, where confirming examples for even considerably larger numbers are so easily adduced. As a matter of fact, in playing dominoes, for example, we grasp groups of ten to twelve dots with one glance. Indeed, we even assess their number with total immediacy. It must be observed, however, that in such cases we can speak neither of an actual colligating nor of an actual enumerating. The number name is here directly associated with the characteristic sensuous appearance, and is then recalled on each occasion by means of that appearance without any conceptual mediation. With groups that large, as everyone can test, a direct and authentic collection and enumeration is an impossibility. In the case of very small groups, of two or three objects, the matter stands in doubt because the successive apprehensions of the elements could ensue so quickly that they themselves would escape focused attention. Therefore the cautious manner of speaking in the text.
By overlooking this circumstance, the most recent analyst of the number concept, W. Brix, has also fallen into errors in his psychologically untenable attempt to conceive of “the number of temporal intuition, or the cardinal” as the second genetic level in the development of the number concept. The first level is supposed to be Represented [repräsentiert] through “the number of spatial intuition.” (W. Wundt, Philosophische Studien, V, Leipzig 1887, pp. 671ff.) From the possibility of continuing the successive positing of units arbitrarily far, Brix draws the highly precarious conclusion (Ibid., p. 675) that in such a way arbitrarily large numbers can be formed. But the mere succession of the repeated positings does not yet guarantee any synthesis, without which the collective unity of the number is inconceivable. It is precisely upon the inability actually to carry out such a synthesis, as we shall later discuss, that every attempt to form an authentic representation of the higher groups and numbers factually runs aground. Through the simplest of experiments Brix could have convinced himself that even nineteen positings of units are not clearly distinguishable from twenty, unless by the indirect means of symbolizations that serve as surrogates for syntheses actually carried out. But of course without the possibility of such distinction there can be no talk of an actual formation of the numbers concerned in terms of sequences of posited units.
Psychologie als Wissenschaft, Königsberg 1825, Part. II, p. 162.
System der Logik als Kunstlehre des Denkens, Berlin 1842, Part I, p. 279n.
I. Kant, Kritik der reinen Vernunft, Sämmtliche Werke, Hartenstein edition, Vol. III, Leipzig 1867, p. 144. B 180
Ibid, p. 142. B 179-180
A. Bain, Logic, 2nd ed., Part II, London 1873, p. 201f. Cp. the cogent critique of Bain’s theory in Chr. Sigwart, Logik, Vol. II, Freiburg i. Br. 1878, pp. 39ff.
Cf. H. Hankel, Vorlesungen über die complexen Zahlen und ihre Functionen, Part I, Leipzig 1867,p. 17.
Philosophische Aufsätze. Eduard Zeller zu seinem fünfzig-jährigen Doctor-Jubiläum gewidmet, Leipzig 1887.
See the Appendix to Part I of the present work.
Logische Studien, Iserlohn 1877, pp. 140f.
Ibid., p. 141.
Geschichte des Materialismus, Book II, 3rd edition, Iserlohn 1877, p. 26.
See the quotation on page 18 above.
Cp. the quotation following farther below.
Kritik der reinen Vernunft, B, §16 (Hartenstein ed., Vol. III, p. 117. B 134
Ibid., B, §15 (Hartenstein, p. 114). B 130
Cp., besides the above quotations, the Geschichte des Materialismus, Book II, 3rd Edition, pp. 119ff. In the Logischen Studien, pp 135f, he explains that we “at first” would have “in this expression [’synthesis’] little more than a formulation of the fact that in all our representations the unity of that which is manifold is present. …” But a few lines later the fact of synthesis is spoken of as a “process,” “through which we, as subject, first arise.”
Logische Studien, p. 148.
Ibid., p. 147.
Cp. C. Stumpf, Tonpsychologie, Vol. I, Leipzig 1883, pp. 105ff.
The mistakes here censured led Lange to strange results. The origin of the categories is supposed to lie in the representation of space. Its properties form the “norm for the functions of our understanding,” etc. “Thus,” Lange finally sums up (Ibid., p. 149), “the representation of space, along with its properties constitutive of our understanding, is revealed as the enduring and defining primitive form of our mental essence, as the true objective counterpart of our transcendental ego.” As soon as one tries to find a clear sense in these phrases, they disperse into nothingness.
J. J. Baumann, Die Lehre von Raum, Zeit und Mathematik in der neueren Philosophie, Berlin 1869, Vol. II, p. 571.
Ibid., p. 669
Ibid., P. 675.
Ibid., P. 671.
Ibid., p. 670.
Logische Studien, p. 140.
In: W. Wundt, Philos. Studien, Vol. V.
Ibid., pp. 671f.
Ibid., p. 672.
Among mathematicians, P. du Bois-Reymond may be mentioned here. In his Allgemeinen Functionentheorie, Part I, Tübingen 1882, p. 16, he comments that “Number is as it were what is left over in our soul when all that distinguished the things has evaporated and there is retained only the representation that the things were separate.” To be sure, being distinct in the sense of being separate is used here, which is no necessary element in a distinction theory of number. Moreover, du Bois-Reymond does not clearly hold to this idea of separateness; and he also seems to restrict himself to spatial objects.
Erkenntnistheoretische Logik, Bonn 1878, p. 405.
Ibid., p. 410.
The Principles of Science, 2nd Edition, London 1883, p. 156.
Ibid., pp. 158 and 159.
This also is probably connected with the fact that from time to time, especially in the case of physical contents, one uses “being distinct” as synonymous with “being separate” (in intuition), expressions which do not always coincide, as is to be seen from the relation of whole and part. We therefore do not take this to be an essentially new kind of equivocation.
We will call them “primary relations.” The further details on the division here indicated, which will prove to be important for purposes of a characterization of the collective combination, are given in Chapter III.
Chr. Sigwart, Logik, Vol. II, p. 37.
Cp. the quotation above.
Logik, Vol. II, pp. 38f.
Logik, Vol. II, p. 37.
Ibid., p. 38.
Ibid., p. 41.
Ibid., p. 43.
Ibid., p. 36.
Logik, Vol. I, p. 36
Ibid., p. 279n. “The opinion that it is first through distinguishing that a representation becomes determinate forgets that the distinguishing itself is only possible between different representations already present, and that the distinction therefore does not produce the different contents.” Here Sigwart is referring to H. Ulrici, Compendium der Logik, 2nd Edition, Leipzig 1872, p. 60.
Very appropriately, therefore, in his Tonpsychologie (Vol. I, p. 96) Stumpf defines analysis as the noting of a plurality.
Errors with regard to the function of distinguishing in the representation of several objects are so easy to fall into that we are not surprised to find them already present in older writers. Cf. J. Locke, An Essay, Book II, ch. XI, sect. 1, and Book IV, ch. I, sect. 4, and ch. VII, sect. 4, and elsewhere. Further, James Mill, Analysis of the Phenomena of the Human Mind, ed. by J. St. Mill, Vol. II, London 1879, p. 15: “As having a sensation, and a sensation, and knowing them, that is, distinguishing them, are the same thing;.…” He also repeats this assertion many times elsewhere.
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Husserl, E. (2003). Critical Developments. In: Philosophy of Arithmetic. Edmund Husserl, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0060-4_3
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