Σύμβoλoν: An Attempt Toward the Early Origins: Part 2

Introduction

On Symbolism: Further Considerations

On sign and symbol. Following Harré, Lamb and Mecacci (1983), in psychology, a sign is considered to have a symbolic nature when we are not interested in its referential meaning but, rather, in the content borne by it, whose communicative aim, being a psychological factor, could be unconscious. In the psychological context, the symbol should be considered in a different manner to the sign, and the content should be considered in a different way to the referent, the latter being correlated to the sign through the reference relationship, whereas the content is correlated to the symbol through the unconscious component of the former. All that is in agreement with the Jungian theory of symbolism according to which reference and use of signs of direct thought should be ascribed to the action of conscious thought, whereas symbols should be attributed to the joint and inseparable action of conscious and unconscious thought. Moreover, according to these authors, there is a strict connection between the symbolic usage of signs and altered states of consciousness, the latter being seen as more desirable because they allow us to bypass the ordinary vigil (often unpleasant and anxiogenic organization of the Ego. These are often identified, in early adolescence (puberty), as initial psychotic events,2 in alternation with the first normal phases of reasoning, abstraction and hypothesis-making abilities, together with the formation of first feelings of empathy and Ego's decentralization. And all this is in accordance with what has been said above about Ego's splitting. Following Galimberti (2006), Freud himself distinguished between sign and symbol, the former being understood as indicating a more or less direct presence of something, while the latter refers to something which, in turn, may refer to another something achieved only by means of an interpretation. As stated above, displacement and condensation are the main mechanisms through which the

primary process acts. In the first of these,3 the part represents the whole or vice versa, or rather a given idea, object or image is instinctually replaced by another effectively associated with it, even if with modalities that are often not logical. On the other hand, in the second one,4 there is convergence and merging of the drive's cathexes related to different objects or aims. According to Lalli (1993), metaphor and metonymy are the main mechanisms of symbolic activity; at the basis of metaphor, then, lie associations by similitude and analogy. Further, metaphor denotes a thing which is different from the named one, transferring the concept that this thing means out of its usual or normal meaning. Instead, metonymy, in naming a thing or a concept, makes a displacement on the basis of a conceptual relationship. In doing so, the effect is named for the cause, the possessor for the possessed thing, the producer for the product, and, above all, the abstract for the concrete.5 Finally, due to the replacement of a total object with a partial one, in fetishism there is the tendency to replace a part with the whole, albeit the inter-individual relationship is kept, and this principle of the method, as already said, might be, in some respects, compared with the inductive method. On the other hand, following Piscicelli (1994), the displacement of a desire upon a generic image is one of the main semantic mechanisms of signification widely involved in symbolic formation. Through fetish formation, the transfer of the Ego upon the object fetish takes place in degenerative cases, or upon symbolic formations in normal cases (see also above Khan Masud's ideas on perversions).

Again on Symbolism

Following Rogers (1978), the consideration of language in relation to primary and secondary psychic processes requires a rethinking of the theory of symbolism. For instance, to rectify that sort of one-sidedness of symbolism (as in the 1916 Ernest Jones work on the theory of symbolism), Rycroft (1968b) argues against this unilateral view that sexual symbolism belongs solely to the primary process and occurs only by virtue of repression, in dreams. Rycroft (1968a, 1968b) and Beres (1950) instead assume symbolism to be a general ability of mind which is based on perception and which may be used both by the primary process and by the secondary process. According to Rogers (1978), a literary symbol or a piece of symbolic behaviour can, and often will, reflect both primary and secondary processes functioning simultaneously, analogously to the action of a bi-logical process as formulated by I. Matte Blanco (see Iurato, 2013). Symbolism is a ubiquitous process present in all human activity. Following Petocz (2004), amongst the diversity of meanings of symbol, Whitehead (1927, p. 60) highlighted the mystical character that it had, commenting on “a certain unstable mixture of attraction and repulsion” in our attitude towards symbolism. He states that symbolism, from sense presentation to physical bodies, is the most natural and widespread of all symbolic modes. Langer (1942) states that symbolization is the essential act of thought: the symbol-making function is one of man's primary activities, like eating, looking, or moving about. It is the fundamental process of mind, and goes on all the time. The human brain function is constantly carrying out a process of symbolic transformation of experimental data that have come to it. The symbolization is the most natural outcome of how the human mind has transformed that primary need to express oneself.

A Comparison with Autism and a Phylogenetic View

The relationships between creative thought and the spectrum of psychotic disorders have been known since Freud's work. For instance, many valuable studies have been pursued in this direction and, in this regard, the communication of Carapezza and Cuccio (2010) is a good essay on the chief studies (see references therein6) concerning some aspects of the creative abilities of individuals who are shown to have Autistic Spectrum Disorders7 (ASDs). For instance, it turns out that many subjects with ASDs have a peculiar characteristic, called weak central coherence, which means that such subjects have a preference for details without being able to lay out them into a coherent and unitary frame.8 It seems that such a strong preference for and attention to details are also typical characteristics of creative thought, which, in turn, would derive both from a sensorial hypersensibility and from a certain tendency towards a hypersystematization thanks to which they are able to identify structural regularities and symmetries. On the other hand, these last considerations might have certain common points with what we have already said above if one considers the disavowal mechanism as strictly correlated with a sort of search for a penis where it is missing, seen therefore as a tendency towards an anatomic detail as well as the result of a lack of regularity which stems from a comparative examination of sexual gender differences. What is said above regarding non-adult subjects with ASDs, who also have an uncommonly high intellectual level,9 could be interpreted as the extreme outcome of a drastic and emotively poorly managed splitting of the Ego which, nevertheless, does not degenerate into paraphilia because of a hypersymbolization as compensation for fetishistic tendencies. From a phylogenetic viewpoint, symbolism embeds its roots in the mists of time. Indeed, symbols seem to be a psychic legacy dating back to the archaic origins of human beings. The history of religions, folklores and mythologies provides interesting sources connected to the first primitive initiation rites: in this regard, see Ferrero , Marchesini , Valeri , Màdera and Eliade . In particular, very close relationships exist between and fetishism; in this regard see Eliade , Abraham and Casonato . Furthermore, fetishism also has deep historical roots in ethnic-social-anthropological and linguistic contexts, from which it has gradually earned its preeminent

On Mathematical Symbolism, Creativity and Other

On mathematical symbolism. There have been several authors who have treated the relationships between exact and natural sciences and psychoanalysis: amongst them,14 C. G. Jung, C. A. Meier, W. Pauli, I. Matte Blanco, E. Von Domarus, S. Ferenczi, I. Hermann, J. Lacan, M. Klein and W. R. Bion. According to Canestri and Oliva (1991),

who analysed the inhibitions in second degree school mathematics, it is possible to speak of a certain “mathematical anxiety” felt by those who use mathematical formulas, like

for every a, b ∈ ℤ and p ∈ ℤ, which refers to a kind of trauma due to the emotive impact with something whose existence, that is the of the formula, is misunderstood; and this, notwithstanding the fact that they showed a certain desire towards mathematics. To find such a meaning, the student is forced to compare herself or himself with an “exterior”, that is the mathematical reality, which has its specific meaningful organizations. For her or him, such a meaning hasn't been symbolically included into the formulas but rather it has fully and defensively taken the place of the formula itself. Thus, we descry certain analogies with aspects of disavowal. In their important paper, Canestri and Oliva also analysed some youngsters suffering from disorders belonging to ∩ (≠ ø), detecting a common feature, that is to say, their inability to make suitable links and connections, as well as to integrate the various senses, the latter being a characteristic identified by D. Meltzer in his studies on autism. The authors, in analysing their case studies, observed as the usual processes followed in solving more or less correctly certain mathematical questions were characterized by sequentiality and continuity laws in turn inferred by perceptive rules. If one asks them the rationale behind these procedures, they are unable to give a correct answer: simply, for them, an algebraic equation is true or false depending on whether they consider a mathematical formula as a (we say, the missing female penis). Canestri and Oliva hence observe that this fact seems to refer to the H. Segal of psychotic thought that has been recalled in the first part of this paper. Canestri and Oliva repeatedly refer toof cases in which the students feel inadequacy, shame and guilt senses in front of the question of establishing the truth or falsity of an algebraic equation. It is as if the notion of “equation with its resolution” was deposited or stored in some part of the mind (unconscious) and is re-evocable only by means of well-determined perceptive stimuli. In learning mathematics, it is necessary to refrain from the sensual gratification provided by sensorial perception. The mental operations of abstraction and formalization impose the abandonment of the known and concrete for the unknown and abstract, thus entailing a certain tolerance of the (we again say, of the female penis). The authors refer to having the impression that many students do not endure such a sensorial deprivation, trying immediately to rebuild up the lost concrete object but not according to a right mental elaboration as above recalled. They immediately follow the known and well-established rules and institutionalized practices, without being able to autonomously and originally create new connections, possible analogies, comparisons and relationships, above all in the face of a new and complex problem. In conclusion, at the basis of these learning mathematics problems, Canestri and Oliva hypothesize a disorder of the symbolic function very similar to that involved in psychotic disorders: in coping with a mathematical task requiring a high symbolic performance, many students feel an unmanageable mental pain that triggers a specific defence mechanism (we say, the disavowal one) that avoids the pain with the construction of a new surrogating reality which is plausible at the perceptive level but misunderstands the true reality, that is to say, the existence of the meaning of mathematical formulas. What was said by these authors is clearly coherently placeable within the framework here outlined and based on the disavowal mechanism. Again, the authors affirm that the various possible solutions and strategies adopted by students to overcome this sometimes have a “psychotic” nature, other times a “ neurotic” nature. According to them, this “mental pain”, felt in the face of an abstract reasoning, seems to be quite widespread. Canestri and Oliva try to explain these facts through different psychodynamic models, including the Freudian one based on

Further Notes on Mathematical Symbolism

Later on, Canestri and Oliva (1991) re-present their considerations in the light of H. Segal's work on symbolism whose main lines were outlined in the first part of this paper. In discussing his thought, we have referred to a meaningful clinical case in which a schizophrenic patient identified a violin with his penis. This means that he found some element common both to the violin and the penis which has obscured all the others that made these two entities distinct, thus passing to an identification of the whole only on the basis of this primary common element. For instance, such a common element might be the geometric or material – hence perceptive – analogy of form between them, which prevails over all the other possible discriminating elements.18 Therefore, from a perceptive element of equality, the schizophrenic patient has passed to an identification of meaning of the two actions, namely playing a violin and masturbation, thus ignoring every minimal pragmatic contextuality. There is, in short, a substantial lack of abstract elaboration. This is the main feature that identifies, in a given social-cultural context, a

psychotic disorder that, nevertheless, may be considered with a certain degree of acceptability in some social-cultural contexts and, above all, in children. In fact, the latter often identify the name with the (named) thing: for instance, for the child, the name “father” coincides only with her or his physical personal father, i.e. the child is unable to consider the name “father” as having something to do with the relational structure of human society, hence of interpersonal nature.19 In this regard, Canestri and Oliva refer to an analogous case which takes place in consideration of the formula ( + ) which is directly equated with + 2 + instead to give rise to the latter through detailed calculations starting from the former, thereby ignoring the deep relational structure existing between these two entities. These modes of reasoning are coming back to slight forms of a psychotic mechanism that these authors would consider to be present in every human being but not in a psychopathological manner; and this is also in line with I. Matte Blanco's bi-logical process theory. Canestri and Oliva say that such types of mathematical errors are mainly due to wrong thought modalities which resemble psychotic ones. And this is a further confirmation of our main hypothesis of this paper, i.e. to put the disavowal mechanism at the primary basis of abstract thought as well as a general psychic mechanism. Furthermore, the basic work of Canestri and Oliva , besides confirming many points raised in this paper, makes a comparison with other psychological perspectives, such as the cognitive one, highlighting many common points with the psychoanalytic standpoint. In particular, the cognitive perspective also points out the action of certain defence mechanisms to avoid painful stimuli, including the so-called proposed by M. H. Erdelyi . Retaking into account what is said above, the paper of Canestri and Oliva above all stresses a possible similar psychotic mechanism implying a lost object-induced splitting thanks to which it will be possible to give rise to semantic and syntactic structures of abstract thought. Their line of thought is very similar to the one followed in this paper and is based on disavowal mechanisms as well as being coherent with what will be said later about the relations with the bodily image formation. As further confirmation of this pursued line of thought, we report the main points delineated in the very interesting paper by V. H. Rosen . First of all, he states that the concept of number normally arises in connection with certain stages of the maturation of the perception apparatus during the Œdipal period, and this, as well as what will be said, is of fundamental importance for the main arguments that we want to claim here. Indeed, the author goes on to say that in those with a special mathematical gift, it is probable that this maturational sequence takes place at an earlier period in Ego development so that along with the precocious concepts of number and quantity there remain certain archaic Ego defence mechanisms (amongst which we would want to include disavowal) which are later utilized in creative aspects of the (mathematical) process. A large part of the ordinary process of mathematical thought in these gifted individuals is preconscious and utilizes a capacity for decathexis of the conscious perceptual system. The “illumination” experience is a creative act, as is inspiration in other fields, and utilizes the Ego's capacity for controlled regression to unformalized infantile modes of perceiving space and number. It is a highly overdetermined psychic

The patient refers to having suffered pavor nocturnus around the age of four of five. On many occasions, when he was about five years of age, he was suddenly no longer allowed to stay in his mother's bed and the night terrors disappeared shortly after this. Rosen interprets all this by bringing back it to scoptophilic primal scene interests with special reference to his father's erection (the silhouette); hence, the light from the street and the window to direct primal scene curiosity and its replacement by curiosity in the intellectual sphere. This latter is suggested in the reference to the delay in instinctual gratification: “I am considering whether I should masturbate” (p. 130). The dream appears to refer therefore to the process of sublimation and the turning of the night light of the pavor nocturnus into intellectual light. Thereafter, the following two dreams were recounted (p. 134):

I am sitting on the floor and see a snapping turtle through a crack in the door of a room. It is my job to keep the turtle in the room, but it seemed to force its way out despite my vigilance“ and ”I see a small “e” to the “x” power times an equation. I realize that I should factor it out and that ‘e’ to the “x” power is a psychoanalyst which must be taken into account in each factor (p. 134).

These dreams occurred shortly after the episode of sudden illumination of a complex Riemannian geometry problem which employed the mind of the patient. For this purpose, he needed a mathematical book that he wasn't able to find easily where he stayed. A sample copy was available in his hometown so he asked his father to send it as soon as possible. But when he got it he suddenly had an insight into finding a short-cut method of reaching the same result by an original method without even removing the wrapping containing the book. The patient also says that as a child he had thought that all turtles were snapping turtles but was fascinated by them and liked to keep one in a pail. The turtle also reminded him of an individual who retires into his own shell and shuts out the world (like his father). He had noticed that turtles blink in the sun and had associated this with the darkness within their shells and to his own photophobia on exposure to bright light. Turtles can see the outside world while they themselves remain unseen. The crack in the door referred to early experiences of peeping at his sisters. The room recalled his own bedroom at home, and his mother's practice of watching him through a crack in the door when he first went to kindergarten, because of his terror at being left by her. This is what Rosen says about possible interpretations of these dreams. We instead would like to consider a possible alternative interpretative hypothesis according to which a turtle, with

the pulling in and out of his head, might represent a female penis disappearing within the mother body22 like in the Freudian cotton reel game that we will consider later. Then, Rosen also makes a certain interpretation of the second dream, bringing back the e^x to an X-ray examination that the patient has undergone. We would like to propose other possible additional variants. Rosen states that this second dream is presented to indicate the relationship of the illumination experience to the primal scene problem. Later, Rosen says that two themes are referred to which are of importance for the development of the theoretical formulation which will follow: the first refers to the selective use of perception in the service of drive and defence in scoptophilic fantasies, and the second to the narcissistic withdrawal from the real darkness and light surrounding the functional relationships of parental objects and their investment in the neutral symbols and relationships of mathematical invention. We add what follows. If one looks at the geometrical form of the graph of the real function e^x (which was surely known to the patient, given that he was a valid mathematician), it is not possible not to recognize an in-erection phallic symbol. Then, the dream also compares this function with the psychoanalyst who, clearly by transference, corresponds to a father figure, so that this second dream should be more referred to the father's phallus rather than an X-ray examination. Afterwards, Rosen continues to interpret these and other dreams of this patient substantially revolving around the Œdipus complex questions concerning the patient of this interesting case report, to finish with a discussion on mathematical thought on the basis of the previous studies made in this regard. Amongst them, we report both a quotation by E. Kris (an exponent of Ego psychology), namely “that hypercathexis of preconscious mental activity with some quantity of energy withdrawn from the object world to the Ego – from the perceptive system to preconscious thinking – accounts for some of the extraordinary achievements of mentation”, and a hypothesis, related to creative thought and also confirmed by clinical data, according to which there are suggestive borderline features as far as psychosis is concerned; a prominent sensory-motor hypersensitivity is also a predisposing feature. Rosen also noticed the concomitant presence of certain disorders in writing and reading in mathematically gifted subjects, which have their origins during the early latency period with the still unresolved Œdipal conflicts. Thereafter, in discussing the role played by the primary process in creative thought, Rosen reviews many interesting case studies and testimonies amongst which we recall only those relating to N. Lobachevsky's work on non-Euclidean geometry and A. Cayley's work on algebra. Indeed, the former was built on a negation of a sensory reality testing, namely that parallel lines can meet, while the latter is concerned with a particular algebra of matrices which was seen, for many years, as a kind of bizarre algebraic oddity. Such an algebra starts from a postulate, in “paranoiac” fashion, which appears to be an absurd negation of the self-evident, namely that the products of two entities (the matrices) are different depending upon the order in which the multiplication is performed. In both cases, we have a of a fact considered to be evident and, notwithstanding these

On Mathematical Symbolism – Thirdly

The history of mathematics comprises numerous examples and case studies regarding concepts as well as notions having anthropomorphic sources. On the other hand, as remembered by Piaget (1968), anthropomorphism has its roots deep in the history of religions and mythologies, as also briefly recalled above (see also Rosen, 1954; Canestri & Oliva, 1991). Many other studies on the history of mathematics, like those made in Ifrah (1985), confirm the primary role that the human body has played in the origins of the main elementary mathematical concepts, like those of number and order, up to the latest results and examples achieved and outlined by the cognitive science of mathematics as exposed by Lakoff and Núñez (2000). For instance, Georges Ifrah speaks of the “bodily techniques of the number” as regards the archaic origins of human awareness of numeric and ordinal notions, so confirming the anthropological assumptions mentioned above. Furthermore, following Loria (1950), in the history of mathematics, the scholar shouldn't only stop at that epoch in which the human being hadn't yet consciously conceived of abstract numbers. In this period, the individual indicated numbers in a phonetic manner, like, for example, in denoting two sheep, three goats, four oxen and so on, or using proper names for certain objects to denote their parts or their components, like, for instance, in associating the idea (or the function) of one, two, three and five respectively to the word “I”,24 to the wings, to the trefoil, and to the hand. Hence, according to Gino Loria, it would also be interesting to go further back, if one would really like to have some form of protohistoric knowledge on the early origins of numbers. According to Ifrah (1985), the numbers one and two were phylogenetically the first numerical intelligible notions conceived by human beings. These are also the first two numerical notions which are ontogenetically acquired by human beings, so that, in this

Two Further Report Cases and a Freudian Case Study

Herein (and in the next subsection), we report some significant clinical cases drawn from psychoanalytic literature, which partially bear out what is suggested in this paper. Following Dieckmann , it is meaningful to report a clinical case treated by Dieckmann himself29 concerning an eternal student in mathematics who suffers from a form of borderline syndrome. This individual refers to having dreamed that Hitler wasn't dead and that he had retaken power in Germany. He says that the SS had identified him as a Jew due to the fact that he had dark hair and a crooked nose. Some days later, an SS official pulled out a revolver to shoot him in the back of the neck. The consequent strong anxiety woke him. Furthermore, when he described this dream to Dieckmann, he was still full of anxiety, insinuating that he (that is, Dieckmann) was that SS official of the dream, disguised as an analyst, who would have pulled out the revolver to kill him. Clearly, the patient had transferred to him (i.e. Dieckmann), by projective identification, his heavy male aggressiveness (which is a possible symptom of his latent homosexuality which, in turn, is strictly related to the Œdipus complex and to the castration anxiety – see Solomon and Patch (1971). In doing so, the patient hoped that the analyst would free him, with his death, from his strong feelings of anxiety. Dieckmann was quite bewildered by that, but immediately had the strong sensation of being the mother of the patient. With the upper part of his body, Dieckmann started to do some slow movements like dandle himself, uttering some calming sounds without speaking. Therefore, through this projective counteridentification by Dieckmann, the patient became the son of the mother (Dieckmann), and thanks to this process the patient was reassured, and at the end he was able to say: it was just a dream! Now, in this regard, it is clear that the dark hair refers to fetish objects and the crooked nose and the revolver clearly refer to a phallic symbol. Moreover, it is not by chance that the patient was attracted by mathematics. Finally, to further validate what we have covered in this paper, we simply quote two important Freudian works, namely The Schreber Case of 1911, and the 1910 Leonardo da Vinci: A

After various possible (some obvious) interpretations, Freud finished by agreeing with the following one (see Freud, 1989, pp. 152-157, pp. 184-188). First, the curiosity for birds may easily be brought back to the childhood curiosity of Leonardo for childhood sexual explorations. Then, the vulture's tail is, of course, a male genital organ, while the vulture itself is a mythological symbol of maternity. The repeated tail knockings on his mouth mean a strong maternal care to Leonardo, so that he was fascinated and seduced by his mother in childhood. According to Freud, all this firmly engraved on Leonardo curiosity toward genital setting differences, above all in detecting the lack of a female penis, which is at the basis of the strong (sublimated) observational curiosity, also towards nature (mother). The precocious sexual excitation, etched in Leonardo by his mother, has been sublimated in symbolic elaboration which, in turn, was the result of his great childhood curiosity arising from gender sexual differences, in primis by the lack of a female penis (the vulture's tail). Thus, the early sexual researches by Leonardo had a decisive role in his creative attitude. Finally, the possible homosexual tendencies of Leonardo may be explained by the lack of an admonishing paternal figure. On the other hand, this absence leads to not recognizing the order of law, this being also decisive in creatively achieving new results beyond the preconstituted order that determines the borders of the knowledge field of every discipline. All this clearly confirms what is said in this paper, since Leonardo was one of the greatest artists and scientists of all the times.

A Possible Reinterpretation of another Freudian Case Study

As a further case, we also consider, following Vegetti Finzi (1976, pp. 29-32), Freud's second chapter, entitled The Child's Game of Fort-Da, of the well-known 1920 paper Beyond the Pleasure Principle. In it, observing his one and a half-year-old grandson Ernst, Freud tries to interpret the cotton reel game by relating this to the absence of the mother and to fill up the emptiness left by the lost love's object. Whilst the child must undergo the turning away of the mother, with such a game he or she may retaliate against this, becoming an active part of such a relation. It is an essentially verbal game, the words fort [forth] and da [here] symbolically filling up such an object's absence, replacing and

representing it in such a way as to develop the symbolic space, hence the culture. In the cotton reel game, it is possible to identify the three main dimensions of the own lived experience: the real one (the mother30), the imaginary one (the reel) and the symbolic one (the word), overlooking the next Lacanian theory. According to Freud, the throwing out of the object, in such a manner as to disappear, might to be a (symbolic) way to satisfy a repressed drive, consisting in avenging the mother's abandonment, taking the meaning of defiance: namely, the child says, “Fine, begone as well, I do not need you, because I myself am to go away you!” The child himself, one year later, when he was angry with a toy, threw it out, roughly exclaiming: “Go to war!” At that time, indeed, it was told to him that his father was at war. Nevertheless, the child did not feel the father's absence at all, showing that he did not want to be disturbed from his own and exclusive possession of the mother. This is a general way of doing of the children when they desire to express similar aggressive and hostile impulses, hurling out objects instead of persons. As is well known, d'après Melanie Klein, play is the best way to analyse a child's unconscious, and Freud himself started to agree with this perspective with this 1920 work. However, given the age of the child herein considered, this discourse is properly inherent to the passage from the anal to the phallic phase until the Œdipus one, when genitality reaches its apex to guide the whole human personality.31 To this purpose, we would want to add that the above Freudian interpretation of the cotton reel game, especially that regarding the last part involving the father, is perhaps a little too rational (also in consideration of the child's age). Instead, we would want to propose a little modification to this last interpretative part, in accordance with what is done in this paper. To be precise, we wish instead to suggest that this cotton reel game could be related to the lack of a female penis instead of the lack of a mother, also on the basis of what Freud says about the second statement, made around the age of two and a half, by his grandson as regards his own father (who was at war). Maybe, the child's feeling of lack might refer just to this lack of a penis rather than that of a father, given also the simple fact that a reel has a phallic meaning: to be exact, the reel's wire (the phallus) that appears and disappears into the reel (the mother) stands respectively for this penis's presence and absence (or retirement into the mother's body) which, as already said, has strong emotional charges in this period of life. The throwing of an object has the psychoanalytic meaning of a phallic erection; numerous phenomena of this type take place during the transition from the anal to the phallic phase, with an apex in the latter. Nevertheless, the psychoanalytic community has forever been concordant in assuming already existent precocious forms of castration anxiety in males and penis envy in females from two years old (see ) which, however, will change during psychosexual development. As stated above, around two years old, the first steps and words begin, together the incipient formation of own bodily image. Moreover, as we will recall in the next subsections, the first communicative and linguistic functions take place as means and tools to symbolically explain corporal movements or to expressively imitate other external phenomena observed by the child. At this stage, the infant's thought begins to form in concomitance with the first verbal and preverbal representations in turn based on and supported by corporal image formation. The child, at this age, is unable to carry out such complete thought, that is to say, that of the absence of the mother as a real person. Freud himself

From Fetishism to Mathematics and Physics

Now, according to what has been said in a previous section, between two and three years old the phallic phase takes place, in which, as we have seen, the castration complex prevails with related phenomena, in concomitance with the Œdipus complex. As we have seen, in this phase the disavowal mechanism (which starts, in its incipient form, in the preceding anal phase) acts with its consequences, so that the above-mentioned psychic phenomena related to counting and to the evanescence of number three might be correlated with it, or brought back to it, given the Freudian (phallic) psychoanalytic meaning of number three. So, we would want to focus attention just on these last remarks: that is to say, we wish to identify the formation of human symbolic function in concomitance with this Freudian psychosexual evolution phase (namely, the phallic one, but with a view too to the previous anal phase) by means of the disavowal mechanism intended as a fundamental psychic formation mechanism which acts in every human being, but which might have degenerative variants (for instance, toward psychotic disorders or paraphilias). In our view, this basic Freudian mechanism (initially considered as a simple Ego defence mechanism but later hypothesized to be also a normal psychic formation mechanism35 by Freud himself in his last 1938 work) has been quite neglected after Freud, except for some notable exceptions. Furthermore, from what has been said so far, it will be possible to put forward the hypothesis that symbolic thought (including the abstract one) is formed according to this mechanism during this psychic phase (the phallic one). In such a manner, it would also be possible to give some explanation as to what the famous physicist and mathematician Eugene Paul Wigner affirmed (Wigner, 1960) upon the close relationships between mathematics and physics, speaking of an “unreasonable effectiveness of mathematics in the natural sciences” about the efficacy of mathematics to symbolically explain the external phenomenic reality; indeed, according to what has been said in this paper, this could be explained simply by the fact that the

On Conceptual Metaphors

What has been said in this paper about the primary role played by bodily image formation may be also laid out into the general cognitive context of embodied knowledge which, as said above, seems to find experimental confirmation by neuroscience studies on mirror neurons. Within this context the important work of George Lakoff and Rafael E. Núñez on embodied mathematics is therefore laid out. Their celebrated work (Lakoff & Núñez, 2000) is based above all on the general linguistic notion of conceptual metaphor according to which, roughly speaking, abstract human ideas make use of cognitive mechanisms to bring back to sensory-motor experiences; that is to say, the abstract is understood through the concrete. This viewpoint has already been philosophically proposed (see Cuccio, 2012), while a more scientific perspective to this idea will be given by the Greenacre researches. As the authors themselves recall, most of our thoughts and conceptual systems are sources from the cognitive unconscious, which is a construct wider than the psychoanalytic one; indeed, it comprises not only the repressed content but, in general, every sort of non-conscious thought, just to use a simple negation. According to Lakoff and Núñez, perhaps the main and most surprising result achieved by cognitive science is that most of our thoughts are unconscious. And mathematics does not make an exception. All our own ideas start from our corporal experiences, including the mathematical ones. Nevertheless, these authors point out that a deep explanation of the

implicit origins of these ideas is missing. For them, conceptual metaphors are mostly part of the cognitive unconscious. They say that these mainly originate from our childhood experience, invoking a sort of fusion of embodied experiences as sources of them. Such a process is analogous with the formation of corporal image during childhood as we will see later, above all with the notable work of Phyllis Greenacre who conducted fundamental studies on the possible relationships between creativity and perversions. In such a fusion, contemporaneous (or synchronic) active involvement takes place, with emotive attendance, of different parts of the body. But mathematical creativity has nothing to do with elementary arithmetic, but rather with abstraction, which is mainly symbolic in its deep nature. In this regard, the psychoanalytic perspective might accomplish this, as we have tried to do with this paper. Indeed, the disavowal mechanism is basically centred on a corporal insight whose outcomes will then be unconsciously elaborated with repercussions on the Ego structure, hence on cognitive tasks. Therefore, what is here proposed might answer this demand for clarification of these questions that Lakoff and Núñez (whose viewpoint is that of the cognitive science of mathematics) claim. They state that there is an unavoidable unconscious conceptual system underlying the whole mathematical framework. In this regard, the psychoanalytic paradigm may lend a valid model to explain the primary sources of conceptual metaphors. Lakoff and Núñez again make proposal about why mathematical knowledge has not been extended to the unconscious realm. This is not completely true: indeed, the important work of Ignacio Matte Blanco tried to successfully move in this direction, as we have briefly outlined in Iurato . It seems that there is a certain reluctance to use the psychoanalytic pattern where it is needed. We instead assume a certain epistemological viewpoint in the general context of psychological sciences. Indeed, extending an epistemological principle due to the physicist Richard P. Feynman, it is possible to consider each of the various psychological trends as a model describing a part or aspect of the complex psychic reality. Only all together will they give a more unitary and complete view of this entity which is shown to have a multiple and varied nature. So, the psychoanalytic paradigm might shed light upon those unconscious aspects that are not very clear to the cognitive sciences, without demanding full authority. From the point of view outlined in this paper, we have tried to descry a possible origin of symbolic function by action of the disavowal mechanism. This might turn out to be useful to explain many not properly cognitive aspects of mathematical thought, for instance, to (epistemological) integrate the Lakoff and Núñez perspective on the cognitive science of

The reference to the triadic structure40 by such a metaphor notion as well as to its essential meaning to replace or represent an absent object (the female penis in the source domain) with another symbol (in the target domain) is clear. Nevertheless, Hopkins does not make the allusion to the disavowal mechanism and related phenomena either explicit or implicit, even if many interesting points discussed by him might be laid out in this explanatory framework considered in this paper.

On Transitional Objects and Fetishism: I

In Greenacre (1971), the author argues about an important study conducted by her on the nature of inspiration in relation to the phallic phase. At the end of this phase, the child should be able to distinguish herself or himself from the external world. The faeces assume the role of objects which are no longer so strictly correlated to own body but belong to the external world and have the double quality of being good or nasty. In this sense, we are much nearer to the Winnicott notion of transitional object as well as to the notions of good and bad object of Melanie Klein. Thanks to her or his conquered standing position as well as to the sensory-motor refinement, the child is able to distinguish between substance and appearance (or form). The obscurity arouses contrasting sentiments, fascinating or scaring. The faeces have the meaning both of birth

(from the anus) and death. According to Greenacre, in this period, that is around four years old, it seems that there is a correspondence between the first forms of consciousness, good or bad hygienic behaviour and the awareness that dreams are just that and not real events that occurred during the night. The control and usage of physiological needs in this period are closely connected with the internalization and comparison of external inputs with imagination and conscious thought. These control functions are mainly located in the genital area which is highly sensitive to external stimuli. Thereafter, castration anxiety (anal castration) starts to appear, even if it is qualitatively different from the one present in the Œdipal phase (phallic castration). The related modalities of explication are different for males and females. In Greenacre (1971), the author discusses fetish and transitional object from a comparative stance. There exist many common points between fetish and transitional object, the latter practically ubiquitous and not necessarily entailing a degeneration. The latter is usually abandoned after infancy but may be extended to accomplish fetishist functions. The transitional object, as meant by Winnicott, is usually the first object recognized as belonging to the not-Me (or not-I; see above point a) of previous section) but not in a full manner. It is created around one year old from the symbiotic mother-child pair, when verbal capacities are uncertain. Greenacre alludes to a basic role played by the choice and formation of this transitional object in creativity. The fetish is above all a bisexual symbol that allows sexual differences to be refused and affirmed. The fetish is almost never spontaneously abandoned because it is the only one thanks to which a sexual satisfaction is possible. Following Greenacre , by comparing the Winnicott notion of transitional object with that of fetish, it is possible to identify some main common aspects. First, there is a kind of symbolic magic in both. In particular, the symbolic nature of a transitional object is quite primitive, through which the infant starts to cast bridges with the external world even if it initially has a multiform, nebulous, evanescent and changeable nature (like the symbol, as recalled in Part 1) but inspires a sense of confidence and assurance to the possible frustrations and anxieties due to a still weak sense of reality. It is the first object created by the child with the help of the mother, which will assume a meaning assigned to it by the child herself or himself. Through its institution, he or she will be able to establish further object relations. It is ubiquitous, although the period of its usage is variable. Nevertheless, the initial emotional charge cathexed into the transitional object gradually vanishes (often towards games), whereas that of fetish is persistent and continuous also in adult age, without doubts about its nature and existence. In the latter, which should represent the mother phallus, material elements and magic essences (mainly due to enchantments and self-hypnotic procedures) are intimately mixed, in a persistent manner. Second, the relationships with aggressiveness are quite different in both cases. The transitional object rather has supportive, intimate and fond relationships with the mother (from which it arises, as the fetish), not marked by an aggressiveness so intense as to become hostile as in fetishism, which has a coagulated anger sprung out of the fear of castration, mainly due to primal scene traumas. The transitional object undergoes a gradual separation from the mother with a certain amount of aggressiveness, but incomparable with the one that fetish has. It is the cross point between the non-hostile aggressiveness of growth and the object of love. The transitional object usually arises when a good enough is present, otherwise hostile aggressiveness or frustrations will be the predominant features. The fetish has its source from the mother's body, from her genital zone to be precise, hence is characterized by an aggressiveness turned toward an own narcissistic satisfaction and not as a means of love like the transitional object. It takes place in its material and noticeable form through a latency phase to adolescence. Sadomasochistic fantasies and practices are common features of fetishists. The

On Transitional Objects and Fetishism: II

We now discuss the role played by fetish and transitional object in imaginative function, following the careful analysis made by Greenacre (1971). Taking into account what was said in the previous section, the overlapping between transitional object and fetish phenomenologies are non-void, in which their common aspects fall. From this, just when the transitional object gradually starts to vanish, the fetish formation begins almost like a continuation of the former, at least in degeneration cases. According to M. Sperling, Winnicott's transitional object is considered as a kind of infantile fetish, further pointing out the comparison between these entities. At this point, in relation to the non-empty area given by common aspects between fetish (infantile fetish) and transitional object, we might rightly call transitional-fetishist objects the entities falling in this area. Now, we debate on the close relationships existing between transitional-fetishist objects and symbolism, imagination and creativity, also taking into account the fact that during the period of formation of these objects, that is to say from two to four years old, the first language properties start to form. According to Greenacre, the capacity to develop illusion and imagination is the main useful (but also potentially dangerous) feature of a transitional object. In general, the formation of illusion is a general and natural phenomenon that exists throughout life, which may be defined as the consequences of a wrong interpretation of one or more stimuli. When we have a sensorial stimulation, the consequent perception of related data is normally performed. Nevertheless, this sensorial experience is apperceived by the observer according to the emotional state of that moment. The various changes that take place are possible only thanks to rapidity of free associations. In adult age, the illusions are often created in those situations in which there is some obstacle to the usual and normal clearness of sensorial consciousness or apperception, also due to inhibiting or hindering external conditions. Then, the collective situations increase such perturbations and deviations which favour the illusion, above all when a common ideology exists that makes the possible choices biased. In this case, the doubtful or dissenting individual may have a general Weltanschauung that is much more exact or valid than the collective one. In these general terms, it is clear that the child is more inclined to create illusions. The only available means of acquisition for children are those provided by sensorial experiences which are mainly promoted, developed and managed by the mother from pregnancy, where a considerable role is played by the introjection-projection mechanisms of primary identification which are already very active during the prenatal phase (see Sasso, 2007). The ability to distinguish between Me and not-Me, the animate and the inanimate, the mobile and the immobile, and so on, is quite uncertain in the child. The ability to determine and to distinguish between the form and contour of external objects develops with the increasing awareness of the own body acquired thanks to sensations due to endogenous experiences as well as to explorations of the own body. In this precocious period the transitional object appears as a means of comparison between already acquired knowledge and further experimentations, hence as a first object for establishing subsequent non-aggressive and sympathetic object relations in the wake of the previous relationships with the mother and her body of which the transitional object will be the natural continuation beyond the restricted bodily confines towards the wider external world. Gradually, the transitional object will be replaced by a

Fetishism and Bodily Image: I

Thanks to its great plasticity,41 the transitional object will allow further multidimensional connections to be instituted through associations with many other external objects which may be represented by means of a spontaneous playful activity. This activity, which was initially free, spontaneous and casual, gradually becomes constrictive and more organized under the internal pressure of maturational tendencies to give a sense of reality and security in the wake of the good mother aspects, if there are any. The transitional object is the first creative object made by the child, an indispensable help for her or his psychic growth. With her or his own imagination and creativity, the child gives a form to this object, instituting with it various possible relations with other objects of her or his imagination or of the external world. The transitional object is moulded on her or his previous (if any) relationships with the good mother of whom it will constitute her symbolic representative. Instead, it will degenerate into an infantile fetish when the bad mother has instituted a perturbative relation with her child, giving rise to a damaged corporal image which he or she tries to remediate by means of narcissistic satisfactions in the latency phase or puberty, during which the adult fetish starts to form in concomitance with strong castration fears. Starting from these notable Winnicott ideas, in particular considering the transitional object as the first object created by the child, Greenacre was naturally led to consider the obvious links existing between the transitional object formation and creative as well as imaginative abilities. According to the official biographies of the most important scientists in history, there are numerous cases (see, for example, Isaac Newton) in which childish play has played an important role in developing their creative and imaginative abilities. According to Greenacre, illusion is a fundamental human capacity which can be improved by means of the correction of wrong perceptions with the own lived experience that gradually every human being acquires through her or his comparison with the external world. To this end, the transitional object plays a fundamental role in the conformation of the perception with reality, trying to avoid the delirium (or hallucinations) which may arise in the case of inadequacy of the related involved emotional control. In some respects, the relationship between (correct or real) illusion and delirium might be roughly compared to the relationship between transitional object and fetish. However, this situation is rather more complex in early infancy in which a primary role is played by the set of perceptive experiences undergone by the child during her or his development, which take place from the first one-year-old abilities to distinguish the Self from the Other. Nevertheless, in childhood, the child has a sensitivity that is so high that he may perform discriminations between objects that the adult may consider to be equal, this being due to a different higher perception of

Fetishism and Bodily Image: II

In Greenacre (1971), an interesting discussion on a 1969 work by E. Galenson about possible sources of symbolic thought and its influences on sublimation processes is conducted (see also Gay, 1992). Galenson discusses on the verbal, non-verbal processes and their relationships as well as their possible role in creative thought. She stresses verbalized and non-verbalized games and their role in creativity. In this regard, a careful analysis of language function was needed. Play comprises movement, imitation and action and generally it is a spontaneous expression of pleasure and spontaneity. According to K. Groos, play will anticipate the subsequent kinds of adult activity. Play has been a fundamental analysis and research tool upon which the work of Melanie Klein, in psychoanalysis, and of Maria Montessori,45 in pedagogy, has been successfully built up. The various toys and objects involved in play basically have the meaning and the function of prolongation of body parts. They are the result of a kind of cathexis of a surplus of bodily energy whose excess needs to be released during bodily growth, while also taking into account environmental opportunities. The modalities of these energetic investments are multiples, and the plasticity with which they will be satisfied are at the sources of secondary thought processes. It seems that thought is the ability to retain and

to use interior images sprung out from sensory-motor and somatic experiences which are acquired with modalities very similar to those of play, although experimented in different conditions and opportunities. This retention, then, is closely related to the sense of time, which is a valuable support to the forms of representation given to these images. During this period, between the end of the first year and the beginning of the second, communication abilities and language functions begin to develop coherently and parallel to the main formation processes of corporal image, as stated above. The first linguistic phenomena are mainly imitative, like exclamations and preverbal vocalizations to symbolically denote corporal movements. The fact is that language also starts during the period of sphincter development and it is clinically known as well that child language may be greatly influenced by the control of sphincters. Therefore, according to Greenacre, there is a great plasticity in the bodily organization which is the basis for the image of Self, characterized by synchronic or parallel developments of various and different parts of the body which therefore will turn out to influence each other by means of simple energetic displacements. In short, Greenacre thinks that there exists a sort of for symbolism. Furthermore, starting from the above-mentioned discussion on sphincter development, Greenacre points out that the phallic phase plays a fundamental role in the further establishment of object relationships. Finally, Greenacre comes back again to creativity compared with secondary process thought and play. Greenacre points out that certain creative predispositions are innate. Out of these, there is a great running of sensory-motor apparatus which will contribute to a better formation both of the perception field and of the set of possible responses to external stimuli. All this will then be developed and assisted by the symbiotic child-mother relation. The mother at first will constitute the of the child's interests, around whom subsequently other peripheral objects might represent or summarize her body or parts of it, which she will promote in dependence on the degree to which she is a good mother. Greenacre calls these or and these are variously invested by libidinal energy as well as non-hostile aggressive manifestations. Besides being crucial points in the perceptive consciousness field, these substitutes will form the seeds of the subsequent affective relations with the external world of which the mother body is an unavoidable intermediary. These objects will be the first steps through which relationships are interwoven with the external world, toward an increasing own autonomy and independence from the mother. They are also the necessary precursors of play as well as of other creative interests. In conclusion, in the predisposed child, the appearance of multiple symbolic functions is based on the institution of these collective alternatives which are mainly the result of external preverbal or partially verbal impressions entrusted to replace the primary intimate contacts with the mother.

On Algebraic Symbolism and Other

This paper has sprung out of a discussion, exclusively performed within the cognitive psychology context, enjoined with Professor Antonella D'Amico, a valid cognitive psychologist from the Department of Psychology of the University of Palermo, about some gender differences in certain mathematical attitudes observed in children and adolescents.47 To be precise, Lipari and D'Amico (2009) observed major abstraction

ability in boys compared to girls, above all as regards algebraic attitudes in which, as is well known, the symbolic function plays a very crucial role.48 Clearly, from their point of view (that is to say, the cognitive science one), these authors have not given any possible psychoanalytic motivation to this detected fact. The outcomes of Lipari and D'Amico (2009) are also confirmed, again within the cognitive viewpoint, by Real Ortega and Ursini (2010), who detected, however, a certain general difficulty in symbolizing and interpreting algebraic variables, even if males are favoured compared to females. Their following conclusions will be easily laid out into a framework based on the disavowal mechanism. To be precise, their conclusions reported the following:

All the students had more difficulties with the variable as a general number and with related variables than with the variable as a specific unknown. In particular, they had difficulties with the symbolization and interpretation of these two uses of variable. We found gender differences between male and female students in 31 of 33 questions. Males obtained higher scores than females, but these differences were significant only for 12 items. The percentage of unanswered problems was higher for females than males. Significant differences favouring boys were found in the interpretation of a variable as a general number and as a specific unknown, when interpretation, manipulation and symbolization of a variable as a general number were required. Considering the interpretation of a variable in a functional relationship, focusing on the variation of variables and on the range of variation, girls had more difficulties than boys in flexibly moving between different uses of variables. This finding allows us to establish that gender differences exist when students work with two or three uses of variables in the solution procedure. The differences were related to the interpretation of a variable in these three different uses and aspects. Clearly, the gender differences were more significant in those exercises in which students need to shift between different uses of variables as facets of the same mathematical object and when they are required to integrate these different uses. Male and female students tended to interpret

All the above-mentioned authors, then, adduce mainly some social-cultural motivations for trying to explain these differences. In contrast, with this paper, we would like to suggest a first possible psychodynamic explanation for this fact. Indeed, it is clear that the disavowal mechanism has certain substantial different implications in respect of gender differences, due to their intrinsic nature as has been defined and considered above all by Freud but also by post-Freudian authors. Boys have a greater propensity to a symbolic and abstract function than girls because of the different approach with which they emotively experience this female lack of a penis. Hereupon, the action of this general psychic mechanism as well as its outcomes will take place with different modalities. On the other hand, it is now known that many mathematical attitudes get their primary sources mainly from visual-spatial abilities49 as well as from bodily sensations

Further Considerations on Creativity and All That

On Creativity: I

According to Carotenuto (1991), creativity is mainly an Ego's function. Such a creative Ego's function does not exclusively have a concrete character, that is to say, it does not consist uniquely in the production of concrete objects, but instead it may act in various other ways. The aims and objectives of libidinal drives are also shifted toward non-personal objects which Phyllis Greenacre generically called collective alternatives (see above section). Fossi (1983), starting from the previous work of Greenacre51 (1953, 1957, 1971), states that at the basis of creative thought lies the experience of childhood dismay felt by a creative individual, during her or his own infancy, with a very special emotional intensity.52 Such an experience would be related to the vision of a penis and with the features of the Œdipus phase, so that we have another confirmation of what is suggested in this paper. According to Greenacre, the psychosexual development of a creative person is quite different from that of any other normal person, so that the contact points between creative talent and neurotic or psychotic behaviour may exist, that is to say, such an incomplete psychic development cannot exclude predispositions to dissociation phenomena. In creative individuals, certain tendencies toward a precocious mysticism or religious experiences have also been detected, sometimes with identifications with divine figures, and this can again be explained by means of the disavowal mechanism because of the mystic meaning that the fetish may have. The gifted child sublimates strong erotic and aggressive instincts into a “loving affair with the world” of which he or she has an exalted vision, so that the creative thing is meant to be like a loving gift. Creativity expresses an endeavour to harmonize the external world (to which the individual responds in a hypersensitive manner) with the internal one, to gratify the own narcissistic needs and to overcome loneliness. According to Greenacre, the creative process does not use neutralized energy and it does not represent a function of the Ego agency (as stated above) free from conflicts. Instead, the creative mind often undergoes reactive formations which may promote symbolization phenomena. The adult creative individual is subjected

On Creativity: II

Creative thought, with the symbolic function as a result of the disavowal mechanism, has strong instinctive needs that the external reality is not able to satisfy, differently from the fetishist who instead finds, in a partial material object (fetish), a rough satisfaction with them. So, the creative person turns her or his attention towards the fantasy world where he or she finds a surrogate to the satisfaction of her or his own desires. Likewise to neurotics and psychotics, the creative mind is cut out from the reality, but is different from it because he or she may come back to the reality since this is not precluded from him or her. This is explainable through the disavowal mechanism because this is mainly based on the chief fact that a real fact (the awareness of gender sexual difference, namely

the detected lack of a female penis) is, however, always perceived but disavowed, displacing such real but painful perception to the symbolic realm in normal cases, whereas, in degenerate cases, it is displaced to a material fetish object. Again, following Carotenuto (1991), the well-known writer Joseph Conrad seemed to have strong fetishist problems according to a psychoanalytic biography written by Meyer (1967). According to Eissler (1962; 1967), who made some interesting psychoanalytical remarks on Leonardo da Vinci and his work in Meyer (1967), psychopathology is an indispensable element for certain types of higher mental conquests; in this regard, see also Andreasen (2008) and Janka (2004). Finally, as stated above, the remarkable work of Greenacre (1955) is on two creative persons, the writer Jonathan Swift (1667-1745) and the mathematician and writer Lewis Carroll (1832-1898). At the basis of their creative thought, Greenacre identifies a disorder of bodily image, hence forms of fetishism. The whole Part II of the fundamental treatise by Carotenuto (1991) is fully devoted to the creative dimension of human thought, and most of what is reported there is easily explainable through the simple psychodynamic model considered here and mainly based on the disavowal mechanism. All this might turn out to be of a certain usefulness to confirm what we suggest. In particular, we have widely made reference to the remarkable work of Phyllis Greenacre. She began to be interested in perversions and creativity from the 1950s. Her main idea is that fetishism is chiefly the outcome of an imperfect development of the corporal image. She deepened fetishism in relation to Winnicott's transitional object theory, whereupon, Greenacre was naturally led towards creativity: as she herself said, this interest in creativity was due to the influence of Ernst Kris and to her previous work on fetishism. In that period, Greenacre was engaged in studying the celebrated Lewis Carroll work under the strong and stimulating advice of Kris. To be precise, in studying fetishism and similar disorders, she noticed the recurrent presence of sentiments of change of corporal image together with tendencies to identify and personalize different parts of the body. At the same time, Greenacre was also studying the above work of Carroll as well as Jonathan Swift's finding confirmation of what she had noticed, namely that in fairy tales and popular stories there were descriptions similar to the features of fetishism of above, so that she began to study the biographies of these authors, which led her to the basic work . In this way, fetishism and creativity met during the fruitful and notable work of Phyllis Greenacre, a route also followed successfully by Chasseguet-Smirgel .

Conclusions

Finally, we summarize what points of this paper, from our point of view, deserve major attention. On the basis of the fetishism pattern as explained by the last 1938 Freudian thought, and taking into account the disavowal mechanism considered to be, d'après Anna Freud and Laplanche and Pontalis (1973), a general psychic mechanism involved in a basic Ego's splitting which gives rise to two main subagencies (Ideal Ego and Ego's Ideal according to H. Numberg and D. Lagache), it is possible to reach the basis of the first very basic symbolic functions by means of the separation of opposites operated by the dialectic interaction between the above Ego's subagencies.53 To be precise, it is the

dualistic interaction between the Ideal Ego subagency, mainly related to primary narcissism, and the subagency system Ego's Ideal – Super-Ego, mainly related to secondary narcissism, that gives rise to that complex, interrelated and variegated realm of non-conventional and conventional symbols. At the same time, starting from the Freudian assumption of the polymorphous nature of a child (which might justify a kind of ubiquitous nature of childish fetishism in infancy), this Ego's splitting gives rise as well both to symbolic and imaginative elaborations (as in normal cases, marking the passage from nature to culture, including the language) or to degenerations (as in pathological cases of paraphilia). These two alternatives are not completely disjunctive of each other, but always in dialectic interaction between them, with a prevalence of one on the other. It is also possible to understand symbolic or imaginative elaborations in a wider sense, including too neuroses and psychoses, but whose relations with reality are quite different from those related to the remaining normal forms of symbolic elaborations like those involved in natural sciences. As recalled by Laplanche and Pontalis , and as widely mentioned above, the two different psychic attitudes resulting from the Ego's splitting, contrary and independent of each other, are at the foundation of the person's psychoanalytic theory itself. Furthermore, having to do with an intrasystemic Ego's splitting rather than with an agency splitting (for instance between Ego and Id), Freud wished to stress a new psychic mechanism different from repression and negation. Indeed, the main feature of this division process is just that it does not reach the formation of a (synthetic) compromise between these opposite attitudes, but rather it keeps or maintains both simultaneously without establishing between them any dialectic relationships. In doing so, that is to say, in contemporaneously maintaining, at the same level, opposite or contrasting tendencies or attitudes, it will be possible to have that character, unifying and globalizing, that will allow a symbolic function, in accordance with its original etymological meaning. As stated in Part 1, during the passage from the anal phase to the phallic one, gradually reaches its higher value in the sense that, in it, the opposite tendencies lie at the same level, that is, they hold next to one another, this being a characterizing element for symbolic formation. In any case, the symbolic function would be closely related to the outcomes of the disavowal mechanism and then ruled by the various qualitative and quantitative dialectic relationships between these two Ego's sub-agencies, established bit by bit during psychosexual development. In particular, in doing so, it would be possible to provide some psychodynamic motivation to mathematical symbolism in relation to natural science in the cognitive model of G. Lakoff and R.E. Núñez, based on the notion of conceptual metaphor. Furthermore, from what has been said in the paper, it is possible to put forward the hypothesis according to which there may be a gender difference in the formation of bodily image during the first four years of age, which, in turn, reflect different abilities in visual-spatial skills that, as said, are of fundamental importance for mathematical

Acknowledgements

I wish first to thank Professor Antonella D'Amico of the Department of Psychology of the University of Palermo for the very interesting discussions I had with her about cognitive psychology aspects of mathematical thought. I'm also grateful to Professor Mauro La Forgia of the Department of Psychology of the First University of Rome, as well as to Dr. Valentina Cuccio of the Neuroscience Department of the University of Parma, for kindly furnishing me with some of their interesting works. Finally, many thanks to an anonymous referee whose invaluable remarks and hints have certainly been important in improving and correcting many aspects of this paper.

Biography

Giuseppe Iurato was born in Ragusa, Italy, and studied Mathematics and Physics at Bologna, Pisa and Catania, Italy. Presently, he is a PhD student in History and Science Education at Palermo, Italy. His interests mainly regard general history and philosophy of sciences, with particular attention to their foundations and possible relationships.