Abstract
Following ideas elaborated by Hering (Grundzüge der Lehre vom Lichtsinn, Springer, Berlin, 1920) in his celebrated analysis of color, the psychologist and gestalt theorist Otto Selz developed in the 1930s a theory of “natural space”, i.e., space as it is conceived by us. Selz’s thesis is that the geometric laws of natural space describe how the points of this space are related to each other by directions which are ordered in the same way as the points on a sphere. At the end of one of his articles, Selz (Zeitschrift für Psychologie 114:351–362, 1930a, p. 358ff) tries to derive within his framework two of Hilbert’s axioms for Euclidean geometry. Such derivations (if successful) would, according to Selz, disclose the psychological origin and meaning of the geometric axioms and would thus contribute to a clarification of their epistemological status. In the present article Selz’s theory is explained and analyzed, his basic principles are amended, and implicit assumptions are made explicit. It is shown that the resulting system is one of ordered affine geometry in which all of Hilbert’s axioms except the axioms of congruence and those of continuity are derivable. The primary aim of the present paper is to make explicit the basic principles behind Selz’s geometry of natural space. The question of the logical independence of these principle is not investigated.
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Notes
In the winter term 1901 Selz, who else studied in Munich, attended Stumpf’s lectures at the university in Berlin.
Natural space is—as Selz (1930a, p. 362), (1930b, p. 41), (1941a, p. 182) declares—Euclidean. That this is not the case for visual space had been suggested at the time of Selz’s writing, e.g., by the psychologist Johannes von Allesch (1931). The problem of the geometry of visual space is extensively discussed, both from the viewpoint of philosophy and from that of psychology, by Suppes (2002, ch. 6).
This is a technical term of Hering’s theory of visual perception. Thus, for instance, the sun as a “Sehding” is a “flat, circular disc consisting of yellow-red”, a sensation (“Empfindung”) “which he have there where the sun just appears to us”; cf. Hering (1879, p. 345).
Cf. Bischof (1966, pp. 311–315, 327f) for an account of the research on localization which has been relevant for Selz.
The German original has “links von A, vor A […]” which obviously is an oversight since A cannot be to the left or in front of itself.
Thus an explanation for the ego’s ability to determine allocentric location from egocentric data could be that it can imagine to move from its actual position to the allocentric point of reference. Such uses of imagination play a central role in Friedman’s (2000) interpretation of Kant’s philosophy of space and geometry. Given Selz’s critical—though respectful—attitude towards Kant’s philosophy of space (cp. Sect. 2.2), it does not seem probable, however, that he considered his relation of relative position dependent upon such uses of the “Einbildungskraft”. Nevertheless we shall several times make use of such “imaginary travels” in order to illustrate certain concepts and principles of our formal reconstruction of Selz’s theory.
Directions are vector like entities and Fig. 2 reminds of elementary vector algebra and its parallelogram law. Given the close connection between vector spaces and affine geometry, one already here would expect that the geometry of natural space will be affine. Stumpf (1906, p. 28), when discussing his notion of an “immanent structural law” (cf. Sect. 2.2 below), mentions the work of Hermann Grassmann—both his pioneering work on vector algebra and its application to color metrics—in order to support his claim that such laws are amenable to a mathematical treatment. Though there is no hint in Selz’s writing that he was aware of the vector like character of his directions, it is likely that he was accustomed with Stumpf’s treatise since he probably took over the concept of a structural law from it.
Selz’s citation of Kant is not quite precise. In the first edition of the Critique of Pure Reason Kant (1902, IV, p. 84) speaks of the “Gewühl von Erscheinungen”. In Kantian terminology an appearance (“Erscheinung”) is the object of an “empirical intuition” (“empirische Anschauung”), i.e., an intuition which is related to its object by “sensation” (“Empfindung”); cp. Kant (1902, III, p. 50). The same replacement of Erscheiung by Empfindung can be found in the writings of some philosophers from the so-called “Southwest School of Neo-Kantianism” such as Lask (1912, p. 58) and Bauch (1923, p. 259).
“Mais si l’on continuait à admettre la rigueur objective des lois mathématiques, la psychologie d’alors imposait avec l’apparence d’une nécessité logique absolue la solution Kantienne;” Selz (1929, p. 340).
Despite of its non-Kantian orientation, Selz theory of space like that of Kant assigns a distinguished status to “the line” and “the circle”, i.e., to linear and cyclic order; cf. Friedman (2000, p. 189ff) for Kant’s views on this issue.
Axiom 4 corresponds to Halsted’s I.1’. His I.2’ is covered by Axiom 6.1 whereas his I.3’ corresponds to two of our axioms, namely Axioms 5 and 6.2. Finally, the job of his I.4’ is taken over by our Axiom 6.3 in conjunction with Axiom 6.2.—The prime (’) in Halsted’s labels for his axioms is due to the fact that they mirror corresponding axioms in his formulation of standard Euclidean geometry.
Halsted’s (1907, p. 201f) “assumptions of betweeness on the sphere” are (counterparts of) Axs. 7.7 (his II 1’.), 7.8 (II 2’.), 7.2 and 7.3 (II 3’.), 7.9 (II 4’.), 7.4 (II 5’.), 7.5 (II 6.), and 7.10 (II 7’). Axs. 7.1 and 7.6 are missing in Halsted’s system though they are clearly valid given his conception of betweeness as it is explained in the main text. — A well-known alternative to a ternary relation of betweeness is the quaternary relation of separation (“points A and B separate point C from D”) as used in projective geometry, cf. Pieri (1895). Actually this kind of relation has been applied in the informal description of the sphere of local tones in Sect. 2.1 above. The interrelationships between linear order, linear betweeness, cyclic betweeness and separation are studied by Huntington (1935).
A line which enters a triangle through one of its edges but does not pass through any of the three vertices leaves that triangle through one of the other edges; cf. Theorem18.II.4.
The same interchange for planar Euclidean geometry has been made by Pogorelov (1966, pp. 23–25).
Hilbert’s book has gone through thirteen editions with corresponding revisions of the text. Though it is here identified by the year 1899 of the first edition, page references are to the centenary edition from 1999, which is a reprint of the 13th edition from 1987 supplemented by some documentary and historiographic materials. The main text of that edition agrees with that of the 7th edition from 1930, the last one published during Hilbert’s lifetime.
Note with respect to clauses 3, 4, and 5 of Theorem 17 that \(\lnot \mathrm {Coll}(A,B,C)\) implies that the points \(A,\ B\), and C are distinct from each other since \(\mathrm {Coll}(A,B,C)\) is always true if at least two of \(A,\ B\), and C are identical.
I owe the observation that the Euclidean axiom follows form Axiom 12 to an anonymous reviewer of a first version of the present article.
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Acknowledgements
I am grateful to three anonymous referees whose critical comments on a previous version of this article alerted me to some oversights and helped me to make it more readable and more precise. Of course, all remaining shortcomings are my responsibility.
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Robering, K. The Geometry of Otto Selz’s Natural Space. Erkenn 86, 325–354 (2021). https://doi.org/10.1007/s10670-019-00106-5
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DOI: https://doi.org/10.1007/s10670-019-00106-5