Brentano (1976) formulates the theory of boundaries and qualitative continua for a deeper analysis of the nature of the spatial element that is a constitutive part of appearances (1982: 104-115). The primary object of perception is composed of the local determination of the place in the visual field where it occurs and of the quality that fills it. The local determination specifies spatial species that are given “in reality as well as in intuition” as the boundaries of three-dimensional bodies (1982: 111). In this sense, perceiving things depends on perceiving boundaries. The things of ordinary experience are extended and may be connected, because they are closed by boundaries at which they are in contact with one another or from which they are fragmented. The boundaries and the qualitative continuum of space are inseparable concepts. For any two regions of the perceptual world that do not delimit each other, there is a third region between them. The subjects may notice parts through the countless boundaries at which they coincide in the whole thing. A thing is extended in length and breadth but presents a front and a rear side, hence it is a two-dimensional boundary of a body extended in three dimensions (1976: 5).

Brentano defines the boundary as the place in which the parts of things appear co-located and coincide (see Smith, 1988/1989, 1995a). Let two halves of a disc be divided into two symmetric sectors that are uniformly blue and red and consider a line passing through the center. Does it make sense to say that subjects perceive in the boundary line the last blue or the first red point of the sectors? Brentano gives a negative answer. Because of the density of the spatial continuum over which the two colors spread, there is no perceptual reason to single out a blue or a red point as the boundary point of the line. Nor does it make sense to say that no colored point is really perceived because a line is a geometrical object without qualitative properties. In a red disc, the perception of its lines and points implies seeing red. To settle the question, Brentano claims that the line passes through the center at the complex point at which blue and red are co-located, namely at which blue and red points coincide, or equivalently that it is perceived to cross the boundary where the disc ceases to appear blue and begins to appear red. Given this argument, the observation that the subjects see the parts of things from countless boundaries does not imply that a visual thing is composed of an infinite number of elements.

The boundaries are the points and surfaces at which the distinguishable qualities of a continuous closed three-dimensional body coincide.

Brentano (1976: 6) recognizes that perceiving a difference between points or lines depends on sensory thresholds, but he intends the boundaries and the coincidence as phenomenal magnitudes.^{[1]} Suppose that a chessboard is made up of alternating juxtaposed blue and red squares. If each square is divided into four smaller squares, these squares are still perceived. Let the division be continued until smaller and smaller squares are obtained up to the threshold. The individual blue and red squares with their local determinations are no longer visible, yet the subjects still see something: the whole chessboard appears violet, namely as something that participates simultaneously in blue and red. Although no one is able to locate the smallest points belonging either to the blue or to the red squares, they see that the surface presents blue and red in different positions where it appears reddish and bluish, reddish blue or bluish red. They understand that, if possible, mutually delimiting areas of both colors could be found therein.

Indeed, Brentano contends that even the concepts of boundary and continuum derive from experience and that they are different from their mathematical formalization, which overlooks the characteristics that made them suitable to describe perceptual experience (1976: 5-6, 51-52, for the shortcomings of Poincare’s and Dedekind’s theories). The difference is explained by the following example (1976: 52). Imagine a space consisting of a set of spheres rotating at different velocities. Let the two extremes of velocity be 0, namely a sphere at rest, and 1, namely one mile per hour, while other velocities have values obtained by the repeated bisection of the interval [0, 1] starting with the velocity of a third sphere at half a mile per hour. The values yield a mathematical yet not a real continuum of velocity. Instead, imagine a rotating disc in which the velocity is one mile per hour in the farthest outer edge and zero in the center, with all the other values occurring in-between. This is a real continuum because each velocity value is a boundary that subsists only as a part of the series of all values for each point of the rotating disc. The series of values from the outer edge to the center unifies each value in a whole and at the same time each value gives a contribution to the whole by being one of its inner boundaries. On the contrary, in the case of the rotating spheres at different velocities, each velocity value is independent.

A boundary is something that exists only as part of a more extended whole, as it necessarily depends on something else. Moreover, a mathematical boundary can belong to both the inside and the outside of a delimited region, while perceptual boundaries are asymmetrical and belong to what they delimit from the outside. The figure 2 may be an example of what happens when Brentano’s condition is violated. Each polygonal figure with the common boundary pops alternatively out in the foreground if the boundary appears to belong to it, while the other, which accordingly fills the bounded area that lies on the other side of the boundary, appears to slide and to extend a bit behind it.

The belongingness of boundaries to the parts of a continuum yields the distinction between inner and outer boundaries (1976: 5). Any surface obtained by dividing a sphere into two halves is an inner boundary; the surface of the sphere is an outer boundary. Likewise the middle point of a solid sphere is an inner boundary; a point on its surface is an outer boundary. The mutual belongingness underlies another difference from corresponding mathematical concepts. If a continuum consists of boundaries and every boundary exists only in continuity with countless other boundaries, the analysis of the continuum does not require Dedekind’s distinction between density and continuity (Korner and Chisholm, 1988: x-xiii; Smith, 1995a).

This characterization allows a classification of boundaries and continua. If a continuum is a boundary, for example a line or a surface, it exists in connection with other boundaries and continua with a greater number of dimensions (Brentano, 1976: 11b). Since a continuum is re-dimensional if it is composed of varying elements of re species, the classification depends on the number of dimensions (1982: 117ff.). A continuum is one-dimensional if its inner boundaries are only individual points and it falls under the concept of line in the broadest sense. A one-dimensional continuum does not contain continuously extended boundaries but only dimensionless boundary points. A continuum is twodimensional if its inner boundaries are one-dimensional continua and it falls under the concept of surface in the broadest sense. A two-dimensional continuum contains lines as well as points. A continuum is three-dimensional if its inner boundaries are two-dimensional continua and it falls under the concept of space. A thing is a three-dimensional continuum because it is bounded by a surface and each of its parts is separated from other parts by a surface, namely a two-dimensional continuum. Finally, a continuum is re-dimensional if its inner boundaries are continua of re – 1 dimensions.

**Figure 2 ** The Variety of the Phenomenology of Perception

This classification can also outline an analysis of the forms and order of phenomena. Spatial things are necessary continua “per se” For example, there are countless spatial points and lines on the surface of a uniformly red disc which exist only as its inner boundaries. The red of a disc is a continuum “per ac- cidens” due to the surface that it fills. There are also continua that are neither necessary nor per se. In a continuously rising tone, each phase presents a tone that may occur alone. If colors from blue through violet to red vary smoothly and orderly between the edges of a surface, each color can be singled out as a separate point or line. Brentano holds also that any perceptual quality that undergoes increasing or decreasing changes can be described as a continuum. There are greater or lesser degrees of a quality, and between any two degrees there can be still another degree, as in the continuous changes of lightness and saturation of colors or tones. In such cases the distinction between qualities that admit natural extremes and those that do not is fundamental (supra § 2.1.1).

The concepts of primary, simple and multiple continua allow one to account for the phenomenal structures and relations in experience. These concepts are defined on the grounds of the changes a feature allows. A tone that sounds unchanged through time is a simple continuum, a tone whose pitch rises continually is a double continuum, and a tone whose loudness changes smoothly is a per se threefold continuum. Thus a tone is a structure, of which pitch and loudness are necessarily dependent parts. Among multiple conti- nua, a continuum is “primary” if it sets the condition for the continuity of other continua: for instance, the temporal continuum for tonal continua or the spatial continuum for colors continua. The necessary dependence of n-fold con- tinua upon at least (n – i)-folded continua and of all multiple continua upon primary continua accounts for the order in which things, qualities and changes thereof are met in experience.

Boundaries are also a case of phenomenal order. A boundary needs a higher dimensional bearer of the boundary on which it depends, although the bearer is not completely determined by this dependence. Brentano (1933: 65-66) remarks that it is impossible to point out the smallest part of the continuum or its neighbor points upon which the boundary depends. Moreover, the bearer may change while the boundary remains unchanged, or the connection between the boundary and the bearer may be missing on one or more sides. Yet these are not counterexamples for the dependence of boundaries that requires only that the demand of connection with a higher-dimensional bearer be satisfied for at least one of their sides. Brentano (1976: 8) ascribes a direction to boundaries as the side along which they delimit the higher-dimensional whole they belong to. For the asymmetry of boundaries, the outer boundary of a thing is never the boundary of the surrounding space, but it may have different determinations as a function of the number of directions along which it is connected with the rest of the thing (1993: 65f.). Brentano (1976: 8) calls “plerosis” the determination of a boundary along various directions. The plerosis of a point can assume many values because a point may delimit the rest of a thing along many directions. The plerosis of the middle point of a solid sphere is maximal, because it delimits all the possible inner sections of the sphere along every possible direction. If the sphere is divided in two, the analogous point on the plane surface of the hemisphere has half the plerosis of the former because it delimits only half a sphere. Brentano claims that plerosis is a magnitude that admits of more and less because the number of directions change as a continuous manifold. It induces an order onto the boundaries of a thing given that they delimit its parts meeting at a point on a surface, the edge of a disc or the vertex of a cone. The plerosis is a cognate notion of coincidence of boundaries. For example, where a line and a circle tangentially meet, two coincident points lie on the line and the circle and delimit them along either a rectilinear or a curved direction. This does not mean that boundaries are multipliable at will. The initial left boundary point of a line is identical with the initial boundary point of the left to right segment that is obtained by halving the line. In Figure 2 the visual paradox depends on the inconsistency between the full plerosis of the shared contour, if seen as inner boundary, and the half plerosis of the coincident boundaries, if seen as belonging to either surface.

Brentano emphasizes the phenomenological meaning of plerosis when he claims that it is qualitatively determined (1976: 15-16, 51-52). The middle point of a blue disc is the boundary of countless straight and curved blue lines and of all the sectors into which the disc can be divided. If the same surface is instead divided into four white, blue, red and yellow quadrants, then the homologous middle point appears composed of four parts; hence it has a fourth of plerosis of the former, because it delimits only one of the four differently colored sectors which it bounds and is part thereof.

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

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