Bion, W.R., Cogitations. Edited by Francesca Bion. London: Karnac Books (1992)
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I wish now to regard the Oedipus myth, the Sophoclean version of that myth, and Freud’s own discoveries as all being attempts at the resolution of a developmental crux. I hope to show that these attempts at resolution are far more widespread in time, and far more various in the form and method of resolution adopted, than has hitherto been realized or even suspected. One such attempt can be discerned in the issues involved in the production of a scientific deductive system and the calculus that represents it.
With this end in view I propose to examine Euclid’s elements. This brings me back to the point at which I said a search must be made for the intersection of processes analogous to those which we ordinarily see in dreams, with the processes we ordinarily associate with mathematical logic. But before doing this I must give a warning: it is inevitable that I should appear at certain points of the discussion to be seeing sexual, and particularly Oedipal, symbols in certain Euclidean propositions. As I shall make clear, there is nothing new in this: in fact insights date back at least to Plutarch. But in some respects it would be putting the cart before the horse to think that Freud’s Oedipal discoveries elucidate the Euclidean theorem, for I wish to propound the hypothesis that the Euclidean theorem and Freud’s discovery of the Oedipus complex, together with the Oedipus myth and the Sophoclean version of it, are alike in that they are both attempts to resolve the conflicts and problems of which they are at one and the same time the manifestation and the attempted solution. I further wish to show that the problems involved in the interplay between the Positions have also prompted attempts at resolution analogous to those to which I point as associated with the matrix or matrices of the Oedipus myth.
I have previously pointed out that the Oedipus myth itself embodies the problem of knowledge in that a question is posed, demanding an answer. One point on which I shall dwell is, what was the question posed by the Sphinx? To this I shall suggest a solution. I shall also suggest that envy plays a role in the fundamental problem, which has not been appreciated although it is of first importance. This envy is directed against the capacity of the individual that makes him able to negotiate the Positions.
To start with the Sphinx and her question – I stress, her question – tradition has it that the question was: what walks on four feet in the morning, on two at noon, and on three in the evening? And the answer was: man crawls on all fours as a baby, walks upright in the prime of life, and uses a staff in old age.
I shall propose a different answer. But I must first draw attention to an aspect of Euclid’s Fifth Proposition in his first book. We are so accustomed to regard his use of the term, ‘isosceles triangle’, as bearing only the meaning that we associate with a mathematical term, that it is difficult to realize that the Greek term could be translated as, ‘a three-kneed thing with equal legs’. R.B. Onions has shown – and he cannot be accused of any tenderness to Freud’s theories of sexuality – that the knees, in early Greek literature, are very frequently associated with the genitalia [Origins of European Thought, p. 174]. This has made me look at Euclid’s Fifth Proposition in a new light. It also makes one inclined to attempt a revaluation of the question traditionally attributed to the Sphinx. But for the moment I shall rest content with bringing together Euclid 1.5 and the Oedipus myth.
Pp. 200-2
[Undated]
Theorem of Pythagoras
– Euclid 1.47
The side subtending the right angle: the sides containing the right angle. How much can be obtained by ignoring the figure, the diagram, except in so far as it serves a function – like that of the material of a sculpture by Henry Moore – in framing the place where there is no material? To act as a boundary to the open space, that is to say the part where the figure is not.
Then the squares on the sides containing, and the squares on the side subtending, the right angle serve to enclose the triangle – the ‘three-kneed thing’, but also the right angle.
The construction is a trap for light.
Pons Asinorum + Positions
Euclid 1.5 marks the point at which the ‘elements’ of geometry are left behind when the student crosses the Pons.
P.206
[Undated]
The relationship of Ps and D positions to the Oedipus complex
Paranoid-schizoid and depressive positions will from now on be called by me, ‘the Positions’, as a matter of convenience.
The Positions are emotional experiences intrinsic to work. All anxiety is related ultimately to The Anxiety, which has two roots:
(l) the contents of the Oedipal situation, which has as its scientific deductive system (s.d.s.) the Theorem of Pythagoras and as its calculus the relevant portions of algebraic geometry;
(2) fear of the Positions, which have as their s.d.s. the Pons Asinorum, Euclid 1.5, and its associated algebraic calculus.
Both theorems have always engaged the attention of mankind and have been confused with each other by name, e.g. the French call 1.47 the Pons Asinorum (see Heath, Euclid, p. 417).
The 1.47 Oedipus and 1.5 Pons Asinorum belong to different axes in human knowledge, but they intersect: one is related to content of knowledge (1.47 Oedipus), the other to acquisition of knowledge (1.5 Positions). The Oedipal situation may occupy different positions in the hierarchy of hypotheses in different s.d.s.’s. In some it may be an initial formula; in others, a derived formula according to the function which the associated calculus is intended to fulfil.
There must be a counterpart to this in non-scientific discipline, and the same difference in hierarchical positions will be observed.
The s.d.s, associated with Euclid 1.5 and the algebraic calculus associated with it are all associated with the Positions.
The Oedipal situation is associated with Euclid 1.47 and its associated s.d.s. (if different from the theorem itself)) and its associated algebraic calculus.
P. 207
[Undated]
An interpretation
An interpretation should not be given on a single association; a single association is open to an enormous number of interpretations. But two associations, like two points that determine a line, determine an interpretation. Not quite. What do they determine? A direction; a trend of thought perhaps. And does this mean that it is at a point of this kind that one might find the interseсtion or meet a logical product of the psycho-analytical interpretation with the s.d.s. known as Euclidean geometry? Is the statement that two points determine a line really an abstraction from an empirically experienced fact that two ideas, taken together, determine a trend in the speaker’s mind? If so, then it is possible that the s.d.s. that is Euclidean geometry is of great importance in the science of psycho-analysis (Braithwaite, Scientific Explanation, p. 27) and can be used to establish calculi, e.g. algebraic calculi, to ‘fix’ certain psycho-analytic theories and expose fallacies in others, as was the case in mathematics when nineteenth-century mathematicians tried to represent the mathematical ‘proofs’ of certain eighteenth-century mathematicians in an adequate calculus.
I have noticed that something that seems clear when I write it appears to have lost all meaning when I come back to it freshly on a subsequent occasion. The words are all there and are as likely as not clear enough, but I find that I no longer know what they are about. It is like the disappearance, apparently, of all knowledge or understanding of Greek literature in Byzantium, or of astronomical or other ideas that are clearly grasped by one generation but then seem to be completely lost by subsequent generations, e.g. the heliocentric theory of Aristarchus. Is it possible that one of the great functions of abstraction is to ‘fix’ an element, the element, of empirical experience, which without abstraction is lost? Is there something permanent and indestructibly precise about the formula of a scientific deductive system that renders it peculiarly fit for the preservation of ideas? Is this a function of logic – logically necessary propositions? Would a s.d.s. give a permanence to Melanie Klein’s paranoid-schizoid and depressive positions?
How does one find the appropriate calculus? Or invent it if it does not exist? Galileo had to get on without the differential calculus in solving the problem of the freely falling body.
P. 210-1
[Undated]
It has just come to me that the mathematician is one who is capable of α and that it has this consequence: he can see the essential characters in, say, the paranoid-schizoid and depressive positions, or the Oedipal situation, or primal scene, or any other. These situations, or complexes, are seen concretely and, at some point, in visual images. For example, Euclid’s geometrical patterns can easily be visual images that are made abstract. But the mathematician is able to produce abstractions, and signs for these abstractions, and can then replace the abstraction or its sign (reversing the direction of α) with a particular concretization.
If my idea is right, then Euclid could say good-bye to the ‘elements’ (1.5, Elefuga). That is to say, the ‘elements’ could be worked – transformed, in his case – into an abstract visual image (the theorem) and then still further abstracted by Descartes (or rather the use of the system of co-ordinates he worked out) so as to form the whole system of projective geometry, and then abstracted still further to form the system of algebraic geometry. Now the calculus thus elaborated lies ready to use.
Leaving aside the question, for the time being, of how this process of abstraction is to take place and how α affects it, we can see that the calculus that has been produced is available for the solution of all kinds of problems which, on the face of it, are far removed from the basic facts (‘elements’, in this case) from which the original abstraction was effected. But it is probable that it will only serve effectually for the solution of problems in which the concrete facts, which are to be substituted for the variables of the calculus, have a value relative to each other which is similar to the values the original basic facts had relative to each of the other basic original facts. Put in other terms, a Newton explores astronomical space, elaborating his laws of motion in doing so by virtue of an already existing calculus, or calculi, which derived originally by a process of abstraction from the facts of the first space of all – the infant’s space of breast, the Positions (paranoid-schizoid and depressive), Oedipal situation, primal scene. There is, then: a basic pattern, an abstraction, calculi and reapplication of calculi.
A Newton then owes his freedom to investigate to a series of derivations from a basic pattern; but that same pattern also imposes a limitation on his freedom in that only such data tend to be regarded by him as will lend themselves to a scheme fitting in with the original basic pattern. In an extreme form this can lead to such a limitation of selectivity of observed facts that the ‘discovery’, e.g. of the origin of the solar system, is really not much more than a projection onto astronomical facts of a crude and very easily detected emotional problem centring, say, on the primal scene. But in another observer of the stature of Newton, the facts now drawn together by aid of the calculi already in existence can be seen not to be adequately represented by the laws of motion and their representative calculi.
Thus there is produced anew the situation, described by Poincaré, of disparate elements lacking any coherence. Confronted by this repetition of the Positions, again all depends on the individual’s capacity for α. The new Positions can again be negotiated, depending on α-capacity, new visual images, new abstractions, and further calculi, and so on.
In this lies the possibility of belief in the value for scientific enquiry of the original basic patterns, and at the same time an indication of the nature of the limitations from which human intellectual exploration and enquiry is bound to suffer. We can ultimately only see in the universe that which lies within the compass of our mental equipment, and that in turn means that we can only see that which lies within the compass of our ability to solve through α-capacity certain basic elemental positions and patterns, such as the paranoid-schizoid and depressive positions, primal scene and Oedipal situation. All that lies outside those few elementary patterns can only be grasped by virtue of α-capacity which can be brought to bear on a derivation from a derivation from a derivation … to the nth.
Pp. 282-3