The text we are presenting, Mathematics, written about 1920 and published posthumously in 1939 in the fourth volume of Ferenczi’s works (Bausteine zur Psychoanalyse), was translated into English in 1955 by Michael Balint (London: Hogarth Press and Institute of Psychoanalysis) and into Italian in 1974 by Glauco Carloni and Egon Molinari (Guaraldi Editore).

It does not appear as a completed paper, but rather as a series of notes that are original but sometimes enigmatic observations that are not easy to decypher. We have no information on the origins of the theme dealt with in the essay and taken up again a few years later in 1926 in the article: The problem of the affirmation of unpleasure. It is undoubtedly a pioneeristic and unusual theme; even today, essays dealing with logical-mathematical thought from a psychoanalytic point of view are very few in the specialized literature, but we know how intense Ferenczi’s interest was for psychoanalytically investigating all the various products of human intelligence. This cultural ardour was very likely linked to the fact that his father ran the most important book store in Miskolcz (Hungary), a meeting point for intellectuals and artists and an activity that was taken on by his mother after her husband’s death. In 1919, preceding Mathematics, Ferenczi wrote a criticism of the text Civilization and Mechanics by the philosopher and physicist Ernst Mach, who sustains the thesis that automatic machines represent projections into the external world of activities of libido-invested internal organs.

Ferenczi had a tendency to bring back to psychoanalytic investigation specialized debates that were considered external to it, and this seemed to cause Freud a certain amount of dissappointment (M.Stanton, 1990, p.66). This interest can be testified by the fact that one of his dearest and most brilliant scholars, Imre Hermann, carried out a thorough investigation of the principal themes of this article: the problems regarding the origin of talent and logical-mathematical thought through specific texts on the nature and function of logical thought. Among the cultural circumstances that may have favoured Ferenczi’s attention towards mathematics, we may recall the international debate that arose at the beginning of the century regarding the foundations of mathematics, during which (in 1902) the great logician G.Frege, following the criticisms by B.Russel, decided not to publish the second volume of his Principles of Arithmetics, and the contributions by B.Russel and Whitehead in the publication of Principia Matematica (1910). At the beginning of the ‘30s the homonymous groups of logicians and mathematicians were founded in Vienna and Berlin to debate about the foundations of mathematics. These cities were an important cultural reference for Ferenczi and it is possible that even at the time the essay was being drafted he already knew about these problematic discussions.

The key word to a comprehension of the text is the term autosymbolism taken from the Austrian psychoanalyst Silberer, who with this term indicates the phenomenon in which some thought or organic activities (the functioning of an organ) are self-perceived in an unconscious manner and projected into a conscious symbolic representation.

Through repeated self-observation, Silberer had discovered that at the moment of the passage between being awake and going to sleep he produced clear images that could be interpreted as symbolic descriptions of what happened at the moment when his effort to think of a given topic contrasted with his incipient sleepiness. This discovery was very favourably accepted by Freud who had already previously hypothesized the possibility of an "endopsychic perception" - a kind of unconscious internal perception of the functioning of the psychic apparatus not recognized as such, but transferred into the external world as a representative base for superstitious beliefs and the production of myths.

According to Silberer, the typical phenomenon of the ipnagogical phase materialized in concrete visual images (auto-symbolic representations) that he divided into three categories according to whether they referred to thought contents (material phenomena), to the modalities with through the conscious functions (functional phenomena), or, lastly, to the reflection of somatic conditions and processes of any nature or emotional states correlated with all these sensations (somatic phenomena). Some "mixed" representations were also possible. In the following essays he broadened the concept of auto-symbolic phenomenon excessively, to the point that Freud himself, after an initial approval, took his distance due to some theoretical consequences that could not be integrated with psychoanalysis.

Although explicitly referring to Silberer’s functional phenomena, Ferenczi again transforms the concept that oversteps the boundaries of the ipnagogic situation to become the founding and driving element for the relationship between the individual and the external world. According to the Hungarian psychoanalyst, the individual goes through "stages of development in his conquest of reality" continuously alternating between projective and introjective activities. The auto-symbolic phenomena, as well as representing our internal unconscious world, also have an instrumental function in "dealing with" the external world once repression and negation have been abandoned. The relationship with the latter in fact implies "calculating" between a major and a minor unpleasure. This calculation, regulated by auto-symbolic phenomena, generates and structures the progression of the subsequent phases of development, and the agent that is strengthened and structured by the calculation is the "ego" itself. According to Ferenczi, even the simplest thought action is based on an indefinite number of unconscious operations of calculus in which presumably every type of arithmetical simplification is used (algebra, differential calculus), and thought, by expressing itself in the form of verbal symbols, represents the last integration of this complex ability for calculation from which, as in calculating machines, only the final result appears.

Regarding the origin of mathematics, or at least of arithmetic, Ferenczi believes that it consists in the unaware perception of the functioning of the various physiological apparatus responsible for sensorial perception. Thus, the operations of addition and subtraction of the sensorial stimuli at the cellular and neuronal level (that give rise to the perception of the stimulus itself: intensity, form, color, etc.), considering only and specifically their "formal" aspect, would give rise to a symbolic production of the mind as the first elementary part of mathematics, i.e. arithmetic. Similarily, certain traditional sub-divisions of the discipline such as arithmetic, algebra or logic, would become psychic projections of the developmental stages that, starging from the perceptive functions of addition and subtraction of the stimuli typical of the sensory organs, lead through progressive abstractions to symbolisation, conceptualisation, and finally to the formulation of conscious thought.

A proof to this thesis are the so-called "idiot savant", people who often have a low intelligence but manage to do exceptionally complex calculations (square roots, individuation of primary numbers, etc.) without having any idea of the reasoning behind what they are calculating and without knowing the algorithms that lead to the desired results.

Although the author’s main thesis that mathematics "is interior self-perception of one’s own conscious functioning" of which only the formal aspects are grasped, belongs in it’s simplicity to the archeology of the topic, it can in any case be included in the present debate about the nature of mathematics and its discoveries.

As is well known, the universality of some of the mathematical rules (eg. the Pythagoras’ theorem, discovered in the same form by populations who were far apart and of different historical and cultural levels), poses the everlasting problem about the nature of the discipline itself. Is mathematics an invention of the mind that, being common to all humanity, sooner or later produces the same results; or does mathematics form a world of its own, with rules that are independent of common reality (e.g. the discover of non-Euclidean geometry), to which people with particular capacities can accede in order to discover the rules of its functioning?

This is a matter of great actuality today that so far does not appear to have been definitely solved, as is witnessed by the ongoing debate between the two points of view, in which astrophysicists such as Barrow, neuroscientists such as Changeux and mathematicians such as A.Connes take one side or the other.

1. Pcs and cs: Organs for ucs psychic qualities.

Ucs: Organ for physiological qualities (perception-recollection).

Sense organs: Organs for physical qualities.

(Qualities are distinguishable quantities.) 2. (i) Psychic reality is arithmetically

(ii)Physiological reality is algebraically(iii) Ucs reality is symbolically (paralogically)

(Primary process)

4. (iv)Pcs reality is logically (Secondary Individuation
5. (v)Cs (?) reality is process)

measured. Counting machine ‘Comparison with the study

of quantities’ (Mathematics).

3. Contrast between the purely introspective mathematician and the purely extraspective primitive Naturmensch, man of action.

Mathematician-knocked down.

No idea of mathematics-skilful.

Skill demands an immensely precise calculation.

Even a dog can do that.

Thinking animals. Introspection into their own Physis.

4. Pure mathematics is autosymbolism (Silberer).

5. A priori knowledge about the processes of the brain (mind).

* * *

6. Mathematics is instinct.

* * *

7. Mathematician not necessarily clever. (Idiots): peculiar combination (introspection).

* * *

8. Genius: combination of strongly-developed introspection with strong

instincts.

(i) Manifestations of instinctual beings are elaborated by a (syst.) cs capable of combination.

(ii) Genius elaborates his own inst. ‘ideas’.

* * *

9. The problem of mathematical talent. The psycho-analysis of Breuer and Freud hardly occupied itself in the beginning with the problems of ‘talent’. It turned its interest almost exclusively to the changes which the human mind has to undergo after its birth under the influence of the environment. For quite a long while it held itself to be incompetent for research into the constitutional factors, into the innate predispositions and faculties. It was in its beginnings a science centred on therapy and practice, and it was naturally concerned above all with the pathological changes of mental life which, acquired during life, were to be reversed by the analyst's efforts, while it was not able to do much therapeutically with congenital constitution. This first ‘traumatic-cathartic’ period in psycho-analysis was a wholesome reaction against the pre-analytic psychiatry and psychology, which completely withdrew from the investigation of the qualities acquired during life, and tended to explain everything mental by the formula of ‘innate disposition’, and everything psychiatric by that of ‘degeneration’.

1O. The second great epoch in psycho-analysis is connected solely with the name of Freud and deserves to be called a ‘theory of libido’. For the first time, the following-up of the developmental phases from birth to involution of one instinct, that of sexuality, was accomplished, and it became possible to describe all its possible issues and to trace them to their causes. Psycho-analysis could no longer confine itself in this phase to pathology. In order to understand pathology better it had to pay attention to the phenomena of the normal psyche, or those which lay in the borderland between normal and pathological (dreams, jokes, slips, criminality), to the products of the group-mind (myths, folklore, religion, languages), to art, philosophy, science, etc., and had to examine the conditions under which such expressions of the mind arise. But only after the scanty bridging by analytical knowledge of the great gaps in post-natal mental development was it possible for psycho-analysis to extend the connecting links so as to include among the factors also the congenital constitution, this time with good prospect of scientific results. Its material limited it, however - it is true at first somewhat one-sidedly - merely to the discovery of the sexual constitutions and their development; from here, however, some ray of light was thrown on to the sources of other, not sexual, abilities and talents.

11. The third, the present-day, phase of Freud’s psycho-analysis is characterized by metapsychology, this unique construction which, without even the slightest help from anatomy, histology, chemistry, or physics of the nerve substance - tries to divine and build up, purely on the basis of mental analysis, the topical, dynamic, and economic laws (connexions) which govern both the whole mental life and all its normal or abnormal mental acts. In addition, the one- sidedness of the psycho-analytic material was later balanced by the creation of an ego-psychology, based on the study of specific ego-illnesses, and through it the biogenetic parallel was extended to the mind.

It is to be expected that this trend of development in psycho-analysis, which takes into account, in addition to the congenital factor, also the formal and quantitative moments, will further the investigation of the various ‘talents’ which until now have been regarded only as rather more or less of one essentially uncertain (anatomical) ‘disposition’. Such researches, however, are not yet in existence.

I have prefaced my article with this cursory survey of the place of mental talent in psycho-analytical theory in order to demonstrate the narrow foundation on which research into one particular talent has to be built, and the various, often completely heterogeneous, strong-points which would have to be used in such building. The furthering and inhibiting post-natal influences, constitutional factors of the ego, and of sexuality, metapsychological considerations, must all help in motivating a special talent.

12. The foundations on which such a construction could be built up were too narrow and the whole building was too labile and insecure to justify any attempt at systematic investigation. Thus we must content ourselves with trying, armed with the tools of psycho-analytical knowledge, to bring nearer to our understanding one special talent, that of mathematics.

13. (i) Arithmetic = Physics

(ii) Algebra = Physiology (sensory qualities)

(symbol!)

(iii) Higher mathematics

(differential -integral

calculus) = Symbolism

(iv) = Logic

Arithmetic = Physics

Algebra = Physiology

Symbolism = UCS Psychic

Logic = Pcs cs

Progressive abstraction (screening) with the help of phylo- genetically acquired functions.

14. Proof for the reality of the external world.

The introspectively acquired (a priori) mathematical laws prove to be valid also in the ‘external world’.

* * *

15. Man is the sum total of physical, physiological, ucs mental, pcs mental, and cs energy forms.

Inter-relation of physical, physiological and psychological forces during the whole life, perhaps also regressively.

The brain as a counting machine.

Censure a filter. Rheostat.

Sense organs are screens (mathematical).

Mathematical genius is self-observation.

Not objective.

Symbolism is self-observation of the latent ontogenetic disposition.

Logic.

1. Auto-arithmetics: perception of the individual sense

Impressions

 

 

2. Symbolism: summation of sense impressions (amphi-mixis-condensation).
3. Logic: regression to arithmetics at a higher level.

Reckoning with higher units (which as such are condensations).

Reckoning with symbols (notions) is measuring of the symbols (notions).

* * *

16. The Summation of kindred or similar elements is one of the preconditions of the reckoning (counting) function, but is at the same time the preparatory work for association between two ideas, association according to certain definite categories (similarity, simultaneity, equal affective importance, objective or subjective valuation, etc.).

The tendency to associate would be a special expression of the tendency to economize. Thinking is after all only a means of preventing squandering through action. (Trial action with small quantities). When, instead of counting each time with the fingers, one puts a figure as a symbol in the place of a row of figures, much mental effort is saved.

The closer connexion of action based on cautious thinking with the tendency to economize (anal character), and its origin in anal-eroticism, is thus understandable.

17. The fusion of numerous single impressions of the external world into a unit and the combination of this with a symbol is one of the most fundamenta1 phenomena of mental life.

In the ucs the fusions (elementary process) happen according to the principle of similarity (especially similarity of the pleasure tones), in the pcs according to the principle of identity or of equivalence (reality principle).

18. Association is an incomplete fusion of two sensory impressions which therefore means that only part of their contents coincide.

* * *

19. (i) Mathematician: self-observation for the metapsychological process of thought and action.

2. Thinker: trial admission to action ‘under displacement of minimal qualities’.

(iii) Active man, man of action: automatic transformation (performance) of the results of the counting machine into action.

20. (Thinker=transitional type between mathematician and man of action.

‘Sicklied o'er with the pale cast or thought').

21. Mathematician can sense only the formal in the process of intra-psychic excitation.

Thinker: sense for the content of the process or excitation.

Man of action: no interest for it.

(Description of the two types.)

22. Mathematical talent.

Scientific works to date on mathematical talent dealt with:

(i) Phrenological problems on the place of the mathematical sense (Gall, Möbius): third left frontal gyrus which has already to look after so many functions (speech, intelligence, etc.). In contrast to it observations such as those about the great mathematician, Gauss, whose skull was allegedly microcephalous and whose brain-weight was extremely small.

(ii) Connexion between musical and mathematical talent -an obviously secondary question.

3. Much more important are psychiatric observations on the coincidence of great mathematical ability with otherwise strong backwardness of the other kinds of intellectual and moral development, often amounting to imbecility, or even idiocy.

* * *

23. Mathematics =self-observation of one's own cs function.

24. (i) Perceptions work on the pattern of a screening apparatus, unifying the impression of the homogeneous influences of the external world into an algebraic unit.

A total impression in which the individual impressions have been united (?). The individual excitement species are sorted out in this way from the chaos of movements of the external world (already an abstraction).

(ii) These simple perceptions are summed up into a higher algebraic unit: symbol-everything similar connected by some superior common factor.

(iii) The elimination (a further screening) of the differences, another abstraction, a11ows of the formation of conceptions which can stand up to reality testing.

(iv) The relating of these conceptions to each other and the trial admission of action based on them (conception of the consequence)=thinking.

25. The psychologist is in the last instance a self-observer (?) + object, "oscillating" between introspection and observation of objects.

* * *

26. Inhibition as a principle of action already valid with the play of the counting machine. (Protective mechanisms against stimuli.)

27. Utraquism.

A ‘Weltanschauung’, as far as possible faultless, demands a utraquistic attitude (oscillating between introspection and object observation) out of which a reliable reality can be constructed.

* * *

28. The pure logician is the mathematician among the psychologists. He has interest only for the formal in the pcs and projects it into the external world. The psychologist must, in addition to logic, also take into account the sub-intellectual, the ucs ideas and their (fantastic) interplay, also the instincts which are the basis of everything psychic -for it is only the aims of the instincts and their derivatives which form the content of the mind. The psyche is ruled by the tendency to care that the instincts receive satisfaction, and are steered eventually into certain harmless channels which prevent unpleasure, that disturbing external stimuli be removed either by adaptation or by changing the external world, or be diminished as far as possible.

The psychologist must not be the mathematician of the mind, for he must take into account the contents of the mind (which are basically illogical and determined by the instincts).

* * *

29. Logic = mathematics of the pcs

Pleasure principle = mathematics of the ucs

* * *

30. Sense organs are better mathematicians than the ucs (less personal). The pcs tries to repair the errors in calculation committed by the ucs, which is ruled by the pleasure principle.

Insensitive bodies are the most real calculating organs (photography, expressionism).

* * *

31. Screening.

Where this mechanism is present the processes of excitation occasioned by the physiological and psychological stimuli must each time go through a new screening, i.e. being sorted and classified according to quantities. (Light filters in co1our photography:

(1) separation, (2) synthesis.)

Sorting into the memory systems of the ucs Progressive Sorting according to sensory qualities screening ( perc. system)

* * *

32. Conceiving would mean then a function which strives to condense all

these impressions separated into their elements into a unit (conception).

Ucs conception thinks in fantasies, that is to say it integrates the psychic rudiments scattered in the memories according to the rules of association of the ucs, such as high-speed process in time (no absolute timelessness), no subtle contrasts. Principle of similarity.

The principal systems of the ucs: pleasure and unpleasure memory system: overruling; decisive for the eventual fate of the conception. (An attempt to restore the world of objects out of its elements.)

Sense organs separate the world into elements, the mind restores it.

This condensation is a magnificent mathematical achievement. Pcs: independent of pleasure principle, is able to reconstruct reality out of conceptions and to base on it an action suited to the purpose (zweckmässig).

* * *

33. Mathematics of the ucs :

Very primitive, approximative mathematics of similarities, but mathematics nevertheless. Mathematics of the pcs strict.

34. The mathematician must therefore gain insight into the processes of his pcs mental systems, if his results are to correspond with external reality.

(Children's mathematics?) No!

(Idiot's mathematics?)

Arithmetic: addition, subtraction-function of the sense organs.

Algebra : combination of higher order; permutation, geometrical calculations, reckoning with time-function of the ucs. (This is to be supposed also in animals: the dive of the eagle on his prey, the spring of the tiger-demand calculation. Differential and integral calculus, geometrical functions, even though no knowledge of geometry.)

35. Mathematics is a psychic organ projection in the same way that mechanics appears to be a physiological organ projection. Music is likewise a projection outwards of the metapsychological processes which accompany the affective and emotional processes; the congeniality of the musician and the mathematician is based on the same ability for fine self-observation.

36. Sense organs are screening mechanisms for the sorting of a variety of impressions out of the chaotic external world. The first sorting takes place according to certain gross differences in the sense organs which, with the help of their specific protective barriers against stimuli, screen off all the stimuli with the exception of those to which they are sensitive (sight, smell, hearing).

A second screening seems to take place within each sensory field according to certain quantitative relations (particular light qualities according to intensity and colour). Similarly the organ of hearing screens the first fairly chaotic acoustic sense impressions and differentiates between the various noises, and leads to the abstraction of certain acoustic impressions which are pleasurably toned because of their uniformity, as musical tones.

* * *

37. Condensation is the process corresponding to association in the ucs. The condensation into an algebraic unit, the summation, through something common, of numerous separate impressions, which is then the sum total of these separate impressions, can be demonstrated also in the manifest dream elements and in the neurotic symptoms, which are all ‘over-determined’.

38. The work of an acting man is a magnificent condensation performance; the condensed result of a vast quantity of separate calculations and considerations-which may be unnoticed, ucs-gives a result which is the sum total of all these calculations, and this residual sum (result) becomes then capable of being discharged ‘in the residual direction’.

39. The mathematician is a man who has a fine capacity for self-observation of this condensation process, i.e. of the formal, of the ‘functional’ phenomena in Silberer's sense; with this, however, he seems to have spent himself so much that he has no mental energy for performing hardly any of actions. Contrariwise the obviously more objectively orientated man of action.

40. First function of the psychic: separation of the sense impressions into their elements, sorting according to qualities, and according to differences in quantities respectively. Sense impressions are deposited in the memory systems in this form. Keeping control over this vast material demands condensations (tendency to economy and simplification).

41. Such ‘condensations’ (formation of conceptions, of memory images, also speculative processes) are always formed afresh from the split-up material of the ucs, according to the actual aims (fantasies). The ucs fantasies arc thought-forms of the ucs. They conform much more to the pleasure principle, but are, however, already bound in a way by the laws of similarity, of succession, etc. The ucs is therefore not really timeless, only much less dependent on chrono1ogical order than the pcs (ceteris paribus simultaneous memories have also here a better chance of being linked associatively, that is to say, of being condensed). In the same way the ucs is not illogical, only paralogical. After finishing the operation of active thinking (condensation) the connexions created for the moment, so to speak ad usum Delphini, again disintegrate into their categories.

* * *

42. The mathematician appears to have a fine self-observation for the metapsychic (also probably physical) processes and finds formulas for the operation in the mind of the condensation and separation functions, projects them, however, into the external world, and believes that he has learnt through external experience.

A very strong argument against this is the eminently intuitive nature of mathematical talent, its connexion with detachment from reality, even with imbecility.

43. Surprisingly these formulas prove valid as ‘applied’ mathematics also in the physical world (technique). Indeed, an argument for the ‘monism’ of the universe, at least for the identity in essence of the metapsychic with the physical.

44. Query: is mathematics abstraction from external experience? or: a priori knowledge?

Solution of this problem perhaps: self-observationis in itself an inner ‘experience’, out of which mathematics is abstracted. That is, on both sides of the P-system mathematical abstractions do occur. In other words : is mathematics internal or external perception? (Up to the present mathematical knowledge has been considered as abstraction (induction?) from external experience.) Here the tracing back of mathematical knowledge and abstraction to internal (self-) observation is attempted.

* * *

45. It is not improbable that we must think of the screening work of the sense organs as a prototype of what happens in the higher mental field, in the ucs. The sorting according to certain categories characterizes the whole memory system, which constitutes the ucs, according to Freud.

The memory systems of the ucs are still sorted-with regard to (achronic) time, to space, etc.-according to the principle of similarity.

What Freud calls a further over-cathexis by the pcs would only be a renewed screening based on the principle of equivalence or identity (reality principle). Separation of the heterogeneous from that which according to the pleasure tone (or otherwise) is similar: logic.

With the help of ‘verbal residues’.

Cs is after all no system, but a special mental act (?).