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algebraic projective geometry, which occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. The mathematical theory of functions consistently applied to the theory of thinking is probably the main reason that led Bion to choose such mathematical instrument as a system of verifiability for the analytical theories. Thus, the breast becomes a point, the movement of the mouth to the breast a line, the two breasts become two points, the projective identification become a hyperbole: “it is convenient to postulate the existence of a mind represented entirely by points, positions of objects, places where something used to be, or would be at some future date. Objects perceived in space contribute to the transformation of these elements (analogous to ξ) into specific no-things” (Bion*, 1965, p. 106). This new space is comparable to the non-linear mathematical concept of Hilbert space (Chuster, 2018), which is a non-Euclidian space. Bion himself says that his intention in using such model was to do for psychoanalysis the same that non-Euclidian models did for geometry. It gave to geometers freedom of thought. For instance, one can observe the mechanism of projective identification in a geometric system as the movement of a hyperbole throwing objects into the air on a curved trajectory onto a distant target. Such distance (corresponding to degrees of distortion in space) defines the type of transformation, as well as its duration in time (intensity). It is important to emphasize that such observational system always introduce two parameters: time and space. “By using the term hyperbole, I mean to bind the constant conjunction of increasing force of emotion with increasing force of evacuation. It is immaterial to hyperbole what emotion is; but on the emotion will depend whether the hyperbolic expression is idealizing or denigrating” (Bion, 1965*, p.142) Another important aspect of the time-space theory is the occurrence of transformations in cycles. The cycles remain in constant progression under certain conceptions are arrived at that could produce a degree of stability to them (relative repetition). However, at the same time, cycles present the idea of vulnerability and instability as they tend to an inertial point. Bion’s metaphor of the image of trees reflected in a mirror of water, as effected by atmospheric conditions, illustrates the instability of the observational system (turbulence) as well as the conditions of the cycles determined by invariants (1965, p. 47). In other words, the cycles depend on the emotional experience produced by a specific combination of links. In chapter seven Bion wrote: “The point and the line represent visual images which remain invariant under a wide range of conditions. The same is true of the visual images associated with the propositions of Euclid; hence the propositions themselves are communicated over long periods of time and between widely separated cultures and races” (Bion 1965*, p.93).
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