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A child's ability to operate reliably with numerical concepts appears in connection with certain stages in the maturation of the perceptual apparatus during the oedipal period. The ability to exclude purely sensory aspects of data in favour of their quantitative value seems decisive. The process is therefore one of selective countercathexis. Observations of hypersensitive infants with a low threshold of the stimulus barrier suggest that the need to erect secondary protective barriers against excessive external stimulation may result in the precocious maturation of some ego functions. It is suggested that these secondary barriers are precursors of the process of selective countercathexis and by their means an ego process may become available precociously for the development of quantitative conceptualization. In those with a special mathematical gift, it is probable that this maturational sequence takes place at an earlier period of ego development, so that along with the precocious concepts of number there remain archaic defence mechanisms which are later utilized in the creative aspects of the process.
In the case reported, a young mathematician experienced spontaneous mathematical insight while undergoing analysis, in connection with repressed feelings of disappointment and rage around objects with which he had previously had positive libidinal ties. The patient dealt with the disturbing, real situation by negation. The creative act, involving momentary 'controlled regression' (Kris, 1950) to the preconscious level, dealt with the repressed archaic conflict by a continuous substitution of symbols for images. By confronting himself at this juncture with a world which was internally consistent within the framework of pure mathematics, the patient was enabled to avoid facing what was inner and what was outer reality. The conflict was resolved in a satisfying aesthetic experience. In this way the mathematician avoids the ambivalent conflict inherent in all oth