Abstract
In the last chapters we have settled all essential questions bearing upon the understanding of the psychological origination and content of the concepts multiplicity and unity, of the determinate number concepts, as well as of the concepts equal, more and less. Our next task will be to confirm the insight won by resolving the difficulties that have been discovered in these concepts, and that seem to involve us in inextricable contradictions and subtleties.
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Notes
Essay, Book II, ch. 16, sect 1.
Ibid., sect. 2. LE
G. Berkeley, A Treatise Concerning the Principles of Human Knowledge. The works of George Berkeley, collected by A. C. Fraser, Vol. I, Oxford 1871, sect. 12, 13, 118-121 (where Locke is criticized without being mentioned by name).
Book II, ch. 12, § 3, Opera philosophica, Erdmann, p. 238. (I have quoted the translation of the New Essays by Peter Remnant and Jonathan Bennett, New York: Cambridge University Press, 1989, p. 145. DW)
Opera philosophica, Erdmann, p. 435. In the year 1684 he still ranked number among the concepts that are common to several senses. Cp. Erdmann, p. 79.
De corpore, VII, 7, cited from J. J. Baumann, Die Lehren von Raum, Zeit und Mathematik, Vol. I, p. 274.
Cp. G. Frege, Die Grundlagen der Arithmetik, p. 38. The sentences following in the text are taken from the same work, p. 57, where they stand, however, in a different context of thought. But since our concern is precisely with the Ideas [Ideen] expressed there, we insert them here.
Cp. the remark by J. F. Herbart, Psychologie als Wissenschaft, Part II, top page 162.
As is well known, the use of 1 in calculation, which was fairly obvious, already belongs to the pre-scientific period of arithmetic, whereas for the introduction of the 0, presupposing a relatively highly developed arithmetical understanding, the wisdom of the Hindus is to be thanked. As M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. 1, Leipzig 1880, p. 159, observes, the Pythagoreans still held the one not to be a number.
Op. cit., p. 57. LE
J. F. Herbart, Op. cit., Part il, p. 162.
W. Volkmann, Lehrbuch der Psychologie, Vol. II, 3rd Edition., Cöthen 1885, p. 114.
De arte combinatoria, Opera phüosophica, ed. J. E. Erdmann, p. 8.
After these discussions it may seem as if the division of names into abstract and general, upon which, e.g., J. St. Mill (Logic, Book I, ch. II, § 4) places such great weight, is one that is useless because it cannot be implemented. If, however, we define an abstract name as a name of an abstract concept, but a general name as a name of the respective conceptual objects, then in each case it is immediately decidable from the sense of the context whether a name is being used as an abstract or general one, in spite of its equivocal character.
This was one consideration that led us, in cases where we remained within the domain of concrete phenomena, to prefer the names “totality” or “group,” and, where we passed over to general concepts, to choose the names “multiplicity,” etc. (Compare also the “Note” on p. 100 above.)
See J. J. Baumann, Die Lehren von Raum, Zeit und Mathematik, Vol. I, p. 242.
Op. cit., p. 161. In the same sense, Fr. Ueberweg, System der Logik, 5th Edition., Bonn 1882, p. 129, explains: “The numeralia can be understood only on the basis of concept formation, for they presuppose the subsumption of homogeneous objects under the relevant concept.” (Cp. also Ibid., p. 141)
Cp. the quotation on p. 51 above.
Op. cit., p. 50.
That is also why the general terms which are based on the concepts of multiplicity and number (e.g., “group”) for the most part incidentally connote the sameness of the objects grasped together. Still, this is least true in the case of the name “totality,” which in this respect also possesses a certain advantage.
Op. cit., p. 50. LE
J. St. Mill, A System of Logic, Book II, ch. VI, § 3. Compare also W. St. Jevons, The Principles of Science, p. 159; and, further, the citations from Kroman following below, from Unsere Naturerkenntnis, Copenhagen 1883. — This view is, moreover, also widely held among mathematicians.
Logique Algorithmique, Liège/Brussels 1877, p. 33. (Also appearing in Vol. I of the Revue Philosophique.) Delboeuf also stands close to Mill in other respects. A few lines further we read: “Number is the scientific expression of the sensible idea of plurality.”
K. Kroman, Unsere Naturerkenntnis, pp. 104–105. Kroman is substantially influenced by the views of A. Lange. (Cp. Chapter II of the present work.)
Cp. Frege, op. cit., pp. 41ff.
Cp. the quotations on pp. 133-134 of the present work.
A Treatise on Human Nature, Part II, sect. II. In the Green and Grose edition, Vol. I, pp. 337-338.
J. J. Baumann, Die Lehren von Raum, Zeit und Mathematik, Vol. II, pp. 669–670.
For easily understandable reasons mathematicians tend to explain the unit precisely in this sense. Cp. P. du Bois-Reymond, Die allgemeine Functionentheorie, Vol. I, pp. 48 and 49, where the confusion with the unit in the sense of #2 clearly stands out. Cp. also Frege, Op. cit., p. 66.
This projected volume of The Philosophy of Arithmetic was never completed as such. But see Essay III. DW
Cp. the quotation on p. 120.
Cp. the Appendix to Part I, below.
Also the name “unification” and the verb “unify” (to combine into a whole) have arisen in this way.
Principles of Human Knowledge, section 12, emphasis added. Cp. also New Theory of Vision, section 109, Fraser ed., Vol. I, p. 25.
Cp. also Frege, Op. cit., p. 58.
Chr. Sigwart, Logik, Vol. II, p. 42.
Lehrbuch zur Einleitung in die Philosophie, Hamburg and Leipzig 1883, § 118.
Op. cit., p. 186.
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Husserl, E. (2003). Discussions Concerning Unity and Multiplicity. In: Philosophy of Arithmetic. Edmund Husserl, vol 10. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0060-4_9
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