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This chapter will be taken up with the construction of some theories that I have found serviceable. It is intended also to serve as an example of the use of the theory of functions and other ideas I have put forward, and thus to take the place of a summary of the main items in the book.
1 The theory of functions and alpha-function are not a part of psychoanalytic theory. They are working tools for the practising psychoanalyst to ease problems of thinking about something that is unknown.
2 The term “function”, used in the sense of a function of the personality, does not have the meaning it possesses for the mathematician or the mathematical logician, though it has features partaking of the meaning of both. I propose it as a term for use in the practice of psychoanalysis; its full designation, if there is any doubt, is “psychoanalytic function of the personality” but otherwise it should be designated simply “function” and given the sign ψ. Alpha-function is a factor of ψ.
3 The term “factor” is the name of an element of any function. It may be represented by the unsaturated element (ξ) in ψ (ξ) and there must be a realization approximating to it. What the realization is that satisfies it, in the mathematical sense of satisfying the terms
of an equation, is a matter to be determined by the psychoanalytic investigation itself.
4 The theory of functions, and alpha-function in particular, does not diminish or increase existing psychoanalytic theories. In this it differs from the arguments that follow.
5 Melanie Klein has described an aspect of projective identification concerned with the modification of infantile fears; the infant projects a part of its psyche, namely its bad feelings, into a good breast. Thence in due course they are removed and re-introjected. During their sojourn in the good breast they are felt to have been modified in such a way that the object that is re-introjected has become tolerable to the infant's psyche.
6 From the above theory I shall