The Phenomenal Space Continuum

Brentano (1988: ii8ff., 128) agrees with Mach (1905) that phenomenal space is distinguished from the abstract spaces of physics and geometry, but he criti­cizes Mach’s conception of pure spatial sensations. Every physical phenom­enon is individualized by the spatial element that is necessarily connected with the quality; hence, there is no space as an additional entity that contains things, rather there are things whose qualities fill various places in which they are localized (1988: 131). Unlike Kant, Brentano ([1982] 1995: 106) contends that we have no a priori intuition of space, rather we perceive space as the con­tinuum that is formed by the connection of the spatial elements. Since spatial elements are connected according to their re-dimensional boundaries, the con­tinuum provides the qualities with an order for their localization. Therefore, it can be abstractly separated from appearances and studied as a particular perceptual form of connection.

The spatial elements of appearances form the phenomenal spaces of each sensory modality (1925: § 12, 1988: 119b). Each phenomenal space implies a reference to the perceiver. For instance, when one sees a color or a form, the appearances point at the place where one is located because they hold necessarily a relation of distance and direction with it. The subject is implied by the phenomenal space as the place that one occupies therein. Brentano’s account of the necessary reference to this place changed over time. First, he maintained that it is directly perceived as the point of the phenomenal space that is the origin of the distance and direction of appearances (1916). It is like the “point of projection” of the localization field of appearances, the center of the phenomenal space that appears outside and in front of the subject. This notion of projection is different from that used by von Helmholtz (Brentano cites J. Muller; cf. Boring, 1950: 356ff.), because it has an inherent spatial mean­ing given that the point itself is included in the phenomenal space as one of its parts. Later he held instead that the place of the subject is perceived only indi­rectly from the distance and orientation of the whole phenomenal space (1928: § 19, 22, 32-33). However, in both accounts the place of the subject is the point where all spatial relations meet. Indeed, it is not distinguished by an individual qualitative or spatial property except that of sharing the same space with ap­pearances. Brentano says that the place of the subject is not individualized by a specific difference with respect to other places in phenomenal space, like for instance the places of blue and red in color space. Rather it is perceived as a place located somewhere, just there where the relations of distance, direction and orientation between appearances require (1916).

This has important consequences. Notwithstanding the different sensory modality, phenomenal spaces are “homogeneous,” in the sense that color, ton­al and tactile appearances are comparable with respect to their localization. Moreover, these spaces are transposed in a common phenomenal space. For instance, a drone is heard on the left or on the right but also localized in rela­tion to the colored things seen on that side (1916). Furthermore, if there is no individual difference in the place of the subject, each subject perceives that she occupies a place of phenomenal space in the same sense as any other does. There is no reason that another subject cannot have the same perception of her place, because the relations of distances, direction and orientations are repeated in every place with respect to any kind of appearances. The particular nature of the place of the subject is the condition of the “universal” character of the phenomenal space. It can be said, then, that in Brentano’s account the phenomenal space is a function of how the spatial elements of appearances vary in localization with respect to perceivers, whose place is preserved as the transposable point that maintains the relations among and with the perceived things. Therefore, all subjects share the properties of the phenomenal space.

The phenomenal space is finite (1988: 119). If one imagines that the visual or the auditory field is filled completely with qualities, the resulting appearances will still amount to a finite perception. This does not preclude that the non­phenomenal space that falls into perception and whose properties are derived from the continuous change of spatial appearances is actually infinite ([1982] 1995: 121, 1988: 119). This means that everything extended is never perceived at once from all of its sides or aspects, because they have to fill the places of phenomenal space, which are given as parts of a finite whole (this character­istic, which distinguishes Brentano’s theory from phenomenalism, is analyzed in depth by Husserl, 1901, 1907, 1913). On this basis Brentano argues that visual space must be three-dimensional. On the one hand, visual things are closed surfaces with a front face and a rear face, the latter of which is momentarily out of sight, that are consequently seen as two-dimensional boundaries of three-dimensional bodies. On the other hand, it is not true to experience to maintain that we see three-dimensional things as the boundaries of a fourth­dimensional space in the same sense in which the surfaces are boundaries of things ([1976] 1988: 5, [1982] 1995: 120). In this connection, the properties of phenomenal space and of the place of the subjects can be further clarified. The parts of phenomenal space in front and outside of the subjects on which the qualities of things are spread and localized can be described as a plane, but this plane is given only as the boundary of a three-dimensional space (see 1928: §§ 19, 33). Likewise, the place that each subject occupies is the point of refer­ence for direction, distance and orientation, but it is given only as the limit of a three-dimensional shared space. The standpoint is necessarily connected to a higher-dimensional part of phenomenal space.

The theory of qualitative continua accounts for the properties of phenom­enal space that make it an ordered layout for appearances. In particular Bren- tano shows that visual space is an instance of the spatial continuum. Visual space is a necessary continuum per se. For instance, there are countless points and lines in a red surface and their existence depends only upon the fact that they are seen as inner boundaries. This property holds also for the surfaces ([1982] 1995: 119, 120). For this reason, we perceive points and surfaces only as boundaries connected with something three-dimensional ([1976] 1988: 5). Vi­sual space is also a primary continuum, namely it is the bearer of the qualita­tive changes of other continua; hence, it must be uniform ([1976] 1988: 15). This means that while movement is a change of place that can be fast, slow or almost unnoticeable before it stops, the places of a surface filled by a red color provide it with a constant change of localization with successively delimiting points from one end of the surface to the other. To be sure, there can be varying rates of change in the phenomenal space. The lines that are not straight have points that can variously change the direction in which they fill the space. This direction is alternately changing in broken lines and continuously varying for more or less curved lines. Yet the inner delimiting points of these lines change their localization for all the line’s length as uniformly as the points of straight lines do.

A primary continuum that is not merely a boundary has to be “straight,” a property that regards the ordering of parts [1976] 1988: 9, ([1982] 1995: 120). A continuum is “straight” if between any two of its inner boundaries there is a third one. A one-dimensional continuum is straight if between any two of its points there is a third one. Such a continuum is a straight line in the broadest sense. A two-dimensional continuum is straight if between any two inner one-dimensional boundaries that are not part of the same straight line there is a third one. Such a continuum is a plane in the broadest sense of a geometrically flat surface. The section of the phenomenal space outside and in front of the subject over which qualities are spread is a plane in this sense. A three-dimensional continuum is straight if between any two of its two-dimensional boundaries, which are not parts of the same plane, there is a third one. The whole phenomenal space is a straight continuum of three di­mensions. This does not imply that there are no curved lines and surfaces, nor does their existence imply that the space is curved ([1982] 1995: 140). A straight re-dimensional continuum contains continua of at least (re-z)-dimension that can be variously curved. Therefore, in three-dimensional space there are curved surfaces and lines. In a curved line there can be indefinitely many points that fill the space in many different directions between one of its start­ing points and any other point, otherwise every intermediate point would lie exactly between the starting and the end point and the line would be straight.

Brentano (1988: 122) maintains that distances in the phenomenal space are not only qualitatively but also quantitatively determinable, in the same sense in which a tone is twice or three times distant in pitch from another tone. In visual space there are no “natural maxima,” like black and white in color space, which allow us to determine the distance of each color from them along the dimension of brightness. Black and white are “natural limits” in the color con­tinuum because one can say that going beyond them is meaningless. In visual space there is nothing similar, yet the points are coincident with perceptually distinguished boundaries of lines, surfaces or solids; hence, the distances be­tween them are determinable magnitudes ([1982] 1995: 118; see 1928: §§ 12, 13, [1982] 1995: 112, 117, for their absolute or relative nature).

Given this description of phenomenal space, Brentano attempts to find the formal properties of geometry that are consistent with them. Actually he does not resort to geometry to define the phenomenal space; rather, he tries to coor­dinate geometry, that is, the abstract theory of measurement and spatial mag­nitudes (1988: 126), with phenomenology, intended in the narrow sense of the theory of the abstract features of the spatial continuum, which are instanced in perceptual experience. In this connection, he argues that the section of phe­nomenal space that appears in front of the place of the subjects corresponds to a Euclidean flat plane, that is, a planimetric plane. This is solely the bound­ary of the continuum of experienced space that is instead a three-dimensional Euclidean flat space, that is, a “stereometric plane.” Thus the properties of the phenomenal extension that is filled by perceived qualities fall under the same general concept described by the geometry of solids. Brentano holds that the concept of the three-dimensional Euclidean flat space is consistent with the fact that phenomenal space is a straight three-dimensional continuum that cannot be resolved in a point manifold that has a lesser dimension than the whole. This is the reason for its homogeneity and uniformity (1988: 140). It is true that a perpendicular can be erected on a plane. Likewise many lines and surfaces can be isolated in a solid given a particular partition. Yet these are magnitudes that can be thought of as being as arbitrarily small as possible boundaries, hence connected to the other higher dimensions that belong to space. In addition, it can be suggested that Brentano is here concerned with a feature that makes the continuum equivalent to a Desarguesian space, in which nowhere does a straight line connecting any two points depart from the plane (cf. Indow, 1991; Wagner, 2006: 45, 48, 57).

Brentano is aware of the debate on the curvature of space, in which he tries to distinguish between the question of the correctness of the Euclidean postu­late on the sum of the interior angles of a triangle and the question about what the space would look like in which it is not verified. He agrees with Riemann (1867) that the existence of “threefold extended manifolds” is a matter of ex­perience. He refuses the theory of geometry as a dictionary of terms that have different meanings in different interpretations of space (cf. Poincare, 1891: 771). He claims that talking of a straight line in a curved space is actually a wrong use of the meaning of the term (Brentano, 1988: 142). He might have rather agreed with Lobachevsky, who claimed that geometry must not be founded on ideal objects but rather on more tangible objects similar to the objects of perception in order to prove by experience the properties of physical space, which may happen to have different properties than the Euclidean one (see Rodin, 2014: 219ff.). In fact, Brentano rejects the hypothesis that the geometry of the non-Euclidean curved space describes correctly the experienced space. However, Brentano (1988: 141) admits that it is consistent to conceive of a space with an arbitrary number of dimensions, whose three-dimensional boundar­ies, abstractly detached from the higher-fold extended manifold, satisfy the el­liptic or hyperbolic geometry. It is interesting that in this connection he makes a hasty reference to Beltrami, who proved the consistency of non-Euclidean geometries.

The meaning of this reference can only be conjectured. He might have been interested in the function of boundaries in Beltrami models that map the Lobachevsky space in the Euclidean space. In the Lobachevsky space, for a point external to a straight line there are at least two parallel straight lines. Consider, for example, figures 21 and 22.

Let r be a straight line and P a point outside it from which the perpendicu­lar PS is drawn to the point S on r. One may suppose that the plane of these lines falls into our perception at least to a certain extent. Any other point on r may be likewise connected to P, thus for instance drawing the lines PC, PA, PR. Lobachevsky calls “secants” such lines that meet a given straight line pass­ing through one of its points. By choosing on r a closer or a farther point with respect to S, one will have respectively “lower” or “upper” secants. In figure 21, for instance, PA is lower than PC and “upper” with respect to PR. Therefore, it is possible to choose on r a point X that is even farther from S, through which an upper secant PX will pass so that any one of the already drawn upper se­cants will always lie between PS and PX. Now let’s consider the straight line s passing through P and perpendicular to PS. Suppose we draw the straight line t through P that diverges as little as possible from a right angle with respect to PS (the same construction holds for u), as in figure 22.

The straight line t will not meet r at a distance that is reasonably within our observational reach. It might meet r at an infinite distance beyond our obser­vation, or it might not. If t is lower than s, any straight line “upper” with re­spect to t and lower than s is also parallel to r. Therefore, for a point P outside r, (i) there exist either secants that meet r at a given point or straight lines that do not; (ii) there exist at least a right and a left straight line passing through P, each lying on one of either side of PS, like t and u, which divide the secant from the non-secant straight lines with respect to r and are parallel to r (in fact Lobachevsky employed the term “parallel” in a narrower sense to denote these kinds of boundary lines, while he called “non-secants” parallel lines in the usual sense). This space has a constant negative curvature, that is, it is a hyper­bolic space. Beltrami (1868) discovered that this space can be mapped in the inner surface of a Euclidean circle, or equivalently that a disc is a model of the hyperbolic space. In the figure 23 the surface of the disc, namely the area that does not include the boundary points of the circumference, maps the plane in such a way that the points on the disc map points, the chords of the disc, which connect points except for the endpoints on the enclosing boundary, map the lines of the plane.

Beltrami proved that the first four Euclidean postulates still hold in this space. Hence should the Lobachevsky geometry involve a contradiction, so the Euclidean geometry would. This model shows that the consistency of Eu­clidean geometry implies the consistency of hyperbolic geometry (see Klein, 1871; Bonola, 1955). However, the Lobachevsky space admits that a line can be infinitely prolonged, while the model is not infinite. One cannot bring in a dis­tance for segments that is greater than the diameter of the circle. Thus, how is one to draw circles of arbitrary radius from an arbitrary center, according to the third Euclidean postulate, in a model in which lines are finite chords? In gen­eral, how does the Lobachevsky space fit to a bounded spatial region? Beltrami holds, for instance, that the points A and B delimit one segment in the finite disc with radius a in the Euclidean plane R², while the boundary of the circle maps points that are at infinite distance from such points in the plane. If the segment AB is extended at will, a chord results whose boundary points R and Q cross the boundary of the circle. Therefore if a point, say B, is moved toward the circle, the Euclidean distance between A and B will grow without bounds.

If the segment AB is moved toward the boundary, its length will grow alike, while the distance defined as the ratio (AQ/AR)/(BQ/BR) remains invariant. Accordingly, the segments of the model map straight lines of infinite length. If one imagines that the radius a tends to infinity, the chords will be stretched until they can become Euclidean straight lines (in fact, the term “straight line” stands for “geodetics”). If we consider the straight line RQ in the disc, there will be: secant lines meeting RQ, which go from P to the points on the arc that are smaller than 180° delimited by R and Q; non-secant lines (or parallels) like t and u along with all those which go from P to the points on the opposite arc greater than 180° delimited by R and Q; and boundary lines (or “parallels” in Lobachevsky’s narrow sense) such as PR and PQ.

This means that the non-Euclidean plane is included in a region of the Eu­clidean plane whose boundary contains the features of the non-Euclidean space. The boundary is ideally visible for an observer placed in the Euclid­ean space that is equivalent to an external viewpoint from which the non­Euclidean space is seen as something filling its boundary. This function of the boundary is characteristic of other spatial models (Beltrami, 1868/1869).

Were the conjecture true that Brentano’s reference to Beltrami is justified by his interest in the function of the boundaries in the Euclidean models, it would meaningfully contribute to assessing the implications of Brentano’s ac­count. In connection with the coordination between phenomenology and ge­ometry, he excludes the possibility that the space of perceptual experience is correctly described as a curved space, because even a non-Euclidean geometry can be mapped into a model of the Euclidean space that matches the prop­erties of visual space. This hints at the question of the Euclidean mapping of perceptual space (cf. Indow, 1991; Hatfield, 2003). The perceptual space may not have the properties of the Euclidean space, as Rubin showed is sometimes the case, yet this does not mean that its phenomenological description cannot be mapped onto a geometrical model. It seems that one can extend by analogy the argument of the consistency between different geometric spaces and claim that however autonomous the properties may be, their internal consistency is somehow connected with some geometrical properties in a model. What, then, are the implications for the coordination of phenomenology and geometry? In Beltrami models the boundary is perceived from an external viewpoint, so what would be the meaning of its appearances if the subject’s standpoint is part of the phenomenal space itself? The question cannot have an answer on the basis of Brentano’s analysis of visual space, but it regards the perception of spatial features of things, such as rigidity, that are essential for the ordinary perception rather than only for the epistemological intelligibility of geometry (infra § 5.8).

Source: Calì Carmelo (2017), Phenomenology of Perception: Theories and Experimental Evidence, Brill.