Showing posts with label philosophy. Show all posts
Showing posts with label philosophy. Show all posts

Wednesday, April 11, 2012

Jungian Themes in Federico Fellini’s 8½
By Michael Nielson

Twentieth century psychologist Carl Jung’s influence is often felt inside the artistic community do to his psychological examination of art and acknowledgement of primordial archetypes.  However, in film analysis much of his influence is limited to his psychological types as opposed to his theories of art and the archetypes.  These are two aspects I feel film analysis would benefit greatly from acknowledging.  In order to better illustrate this, I will examine Federico Fellini’s , and bring aspects of the film into a clearer light through an examination of both the theories of art and archetypes.  I will begin by giving a brief outline of and present details crucial to an understanding of the narrative.  I will next move onto an introduction of Jung’s theory of art, and through exerts from interviews with Fellini, I will illustrate how the two are interconnected.  Finally, a presentation of Jung’s theory of archetypes will show how, through these archetypes, we can better understand the meaning and symbolism of key scenes in a film.  Together these elements will show the wide ranging impact Carl Jung’s system is able to have in the field of film analysis.

I – An Introduction to [1]
            is often considered to be the magnum opus of Italian filmmaker Federico Fellini.  It tells the story of fictional filmmaker Guido Anselmi(Marcello Mastroianni), a character who appears at least in part to be based upon Fellini himself.  Like Fellini, Guido is in the process of making a film that has not yet been fully developed.  This film will revolve around a hero who leads “a group of people to start a new life after a nuclear holocaust” via a rocket, and interactions with both a Catholic cardinal and a mysterious woman in white are the only other details revealed.[2]  The production of the film continues despite this, leading Guido to endure an intense psychological pressure from both producers as well as various Church officials.  There are numerous sequences involving both past collaborators, like his fellow writer Daumier(Jean Rougeul), who acts as his creative conscious, pressuring Guido to live up to past successes.  There are also potential producers, who level him with questions regarding financial and logistic issues of the film.  Another pressure surrounding the film can be seen in Guido’s relations with the Catholic Church, whose opinion still weighs heavily on him, despite an inner impulse to purge himself of its dogma.
Guido’s personal crisis is not limited to only the film, as it also results from troubles with the opposite sex.  Two women who clearly exemplify this are Luisa(Anouk Aimee), his wife, and Carla(Sandra Milo), his mistress.  Carla is the first to enter the story when Guido greets her arrival via train, and we quickly learn their relationship is shrouded in secrecy.  His wife Luisa, who arrives over halfway through the film, appears very happy to see Guido at first, but this quickly changes when she begins suspecting infidelity.  After she leaves Guido at the screen tests for his new film, he is suddenly approached by a third feminine influence, Claudia(Claudia Cardinale), the woman who he had envisioned playing the mysterious woman in white central to his new film.  She will serve an important function that will be understood more clearly after examining Jung’s theory of archetypes. 
            In addition to the scenes which tell the story of Guido in the here and now, the film also flows seamlessly between his memories and dreams.  Though there are only two memory sequences, they both tell us of events that would have a profound effect on Guido’s psychological development.  The first deals with matriarchal figures in his youth, while the other details his experience being caught by church officials with the prostitute, La Saraghina(Edra Gale).  Guido’s dreams give us insight into the nature of his present crisis, and in many ways will be his salvation in overcoming it.  A significant fantasy occurs when Guido’s anxiety reaches a fever pitch during an encounter between Luisa and Carla, causing Guido to become fully engulfed in a fantasy where the women of his life come together in a harem.  Upon arriving, he is immediately showered with love, but the women soon begin to protest Guido’s guidelines, and he is forced to subdue them with violence, after which they encircle a table and eat dinner.[3]    This circular imagery would carry over into the final fantasy of the film, when Guido finally halts production and announces there is, in fact, no film.  Shortly afterward, Guido brings all the characters of his life together to dance in unison, solving the creative crisis that had been the central theme for the duration of the film.

II –Jung’s Theory of Art and 8½.
            The case for using Jung’s psychology as an approach to analyzing film has all but been overlooked in most circles, often in favor of a more Freudian perspective.[4]  What little literature does exist ignores Jung’s theory of art, and instead focuses on a more well know aspect of his system, the psychological types.  An example of this is seen in Don Fredericksen’s article entitled “Two Aspects of a Jungian Perspective Upon Film: Jung and Freud; The Psychology of Types” where he deals primarily with the “typing of psychological attitudes and functions” and the relation of these attitudes to characters in film.[5]  Jung’s psychological types consist of what he refers to as ‘extraverted’ and ‘introverted’ types.  The extraverted type, who is more focused on fertility than self-preservation, occurs “when orientation by the object predominates in such a way that decisions and actions are determined not by subjective views, but by objective conditions.”[6]  In contrast, the introverted type, who is more focused on self-preservation than fertility, occurs when orientation to the object is achieved through subjectivity and “is therefore oriented by the factor in perception and cognition which responds to the sense stimulus in accordance with the individual’s subjective disposition.”[7]  These two base types are further broken down into two rational functions, identified as thinking and feeling, and two irrational functions, identified as sensation and intuition.[8]  Each individual has one function of primary importance that dominate the other three, and one auxiliary function which is subordinate to this primary function.[9]  The remaining two are relegated to the unconscious nonego, where one takes the role of the inferior function, and the other acts as its auxiliary function.[10] 
While I agree with the conclusions Fredericksen draws regarding the usefulness of analyzing characters by means of these types, I feel if we stop our investigation here and fail to take into account other aspects of Jung’s literature, we are missing important ways his system is relevant to film analysis.  The first is Jung’s theory of art, which focuses primarily on literature, but is able to extend its influence to all of the creative arts.  In “Psychology and Literature”, written in 1930 and revised in 1950, Jung distinguishes two separate types of novel, which he refers to as the psychological and visionary types.  The psychological novel “deals with materials drawn from the realm of human consciousness.”[11]  In novels of this type, the author has a hand in illuminating the material to the point all interpretation of the work is concrete.[12]  In contrast, the visionary novel is drawn from “a primordial experience which surpasses man’s understanding.”[13]  Here, the question of the novel’s meaning and the characters psychology are not fully determined by the author, leaving the interpretation in the hands of the reader.[14]  Jung identifies the ideal reader of the psychological novel as the layman, while the ideal reader of the visionary novel is the psychologist.[15]  This is because the psychological type of novel “takes us away from the psychological study of the work of art, and confronts us with the psychic disposition of the poet himself.”[16]  A psychological inquiry has very little interest in the poet, and instead focuses on the creative vision itself.[17]  Because he feels “the vision represents a deeper and more impressive experience than human passion” it is placed at a higher level of importance.[18]
            Works Jung holds as exemplary of this distinction are Goethe’s Faust I and Faust II.[19]  Faust I (1808) “explains itself; there is nothing that the psychologist can add to it that the poet has not already said in better words.”[20]  This part of the tragedy deals with the traditional legend of Faustus selling his soul to the devil incarnate, Mephistopheles, and details a love affair between Faustus and the character of Gretchen.[21]  In Faust II (1832) however, “nothing is self-explanatory and every verse adds to the reader’s need of an interpretation.”[22]  Much of the tragedy involved with Gretchen has been forgotten, and the plot follows a five act structure, but each act has its own unique theme.[23]
Much of Jung’s theory of art is more fully illuminated by examining its genesis in his 1922 article entitled “On the Relation of Analytic Psychology to Poetry.”  Here many of the ideas regarding the creative vision begin to take shape as he investigates literature via psychological enquiry.  Jung very quickly establishes that, because of the inherent limitations of any science, “only that aspect of art which consists in the process of artistic creation can be subject for psychological study.”[24]  Jung introduces the basic distinction between the two contrasting types of art as well, by referencing them to his psychological types.  What came to be regarded as the psychological novel is connected to the extraverted type, while the visionary novel to the introverted type.[25]  This, however, is not meant to classify the creator of the work with the attitude of the work, as the contrast between Faust I and Faust II illustrates, a person of one type can clearly create a work of the opposing type.[26]
Jung’s view is perhaps best summarized in the conclusion of his exposition into the relationship between psychology and literature, when he claims that “a great work of art is like a dream; for all its apparent obviousness it does not explain itself” and “to grasp its meaning, we must allow it to shape us as it had once shaped” the artist who had the creative vision.[27]  This focus on this creative vision itself I feel makes 8½ an exemplary case of a visionary type of film, because, as John Stubbs describes in “Fellini’s Portrait of the Artist as Creative Problem Solver”, it celebrates this moment of creative breakthrough.[28]  This can be expanded on by examining what Fellini has said about the film and his thought process when constructing it.
In his interviews with Gideon Bachmann there are many examples of Fellini embracing a more open-ended approach to filmmaking.  Fellini speaks of being a guest in his “invented dream world” and needs to maintain “a feeling of curious surprise, a feeling of being a visitor, after all, an outsider, even when I am, at the same time, the mayor, the chief of police, and the alien registration office of this whole invented world.”[29]  This can be seen as directly correlating to Jung’s idea that the meaning of a visionary type of novel transcends the scope of its author.  Another example of this is when Fellini describes himself, during the production of a film, as being forced to “maintain a constant equilibrium…between that which you wanted to do, in other words the film as it presented itself in your imaginative sphere, and the one you are actually making” which reveals itself more fully as the production progresses.[30]  Much of what Fellini has said about his films follows this theme of disconnection between the final cut of a film and himself, and he describes the films as being able to “absorb and reflect the emanations of changing psychological, environmental, meteorological and neurotic configurations just like a living person would do.”[31]  This connection, however, is not limited to only the creative process, and also has a deep connection with the channeling of primordial experiences called archetypes, those collective concepts which Jung cited as characteristic of the visionary type of art.[32]

III – The Archetypes
Jung draws much support for his theory of art from skepticism of the scientific attitude psychology is subordinate to.  He claims that all a scientific examination can do is bring “the work of art into the sphere of general human psychology, where many other things besides art have their origin,” namely the archetypes.[33]  Central to Jung’s theory of archetypes is the distinction between the conscious and the unconscious.  The conscious mind is immediate to us and in many cases is thought by the subject to be the lone psyche.[34]  There also exists a second, unconscious mind, however, which is divided again into our own personal unconscious and our inherited collective unconscious.
While the personal unconscious is made up essentially of contents which have at one time been conscious but which have disappeared from consciousness through having been forgotten or repressed, the contents of the collective unconscious have never been in consciousness, and therefore have never been individually acquired, but owe their existence exclusively to heredity.  Whereas the personal unconscious consists for the most part of complexes, the content of the collective unconscious is made up of archetypes.[35]
Jung describes the process of creating visionary art as unconsciously activating these archetypical images and shaping them into the work of art.[36]  These archetypes “stir us because they summon up a voice that is stronger than our own” and transmute “our personal destiny into the destiny of mankind,” evoking “in us all those beneficent forces that ever and anon have enabled humanity to find a refuge from every peril and to outlive the longest night.”[37]  These archetypes indicate the existence of definite forms in the psyche which seem to be “present always and everywhere” as patterns of instinctual behavior that answer the “omnipresent human need” they reflect.[38] 
Due to the extensive number of archetypes Jung referred to, I will focus on the terms key to the examination of : the shadow, persona, syzygy and, finally, the self.  The shadow is introduced as “a moral problem that challenges the whole ego-personality, for no one can become conscious of the shadow without considerable moral effort.  To become conscious of it involves recognizing the dark aspects of the personality as present and real.”[39]  The persona is the aspect of ourselves that we present to the world in an attempt to hide our shadow, and though it begins as an archetype, by the time it reaches peak growth it will be “the part of us most distance from the collective unconscious.”[40]  The anima and the animus, collectively referred to by Jung as the syzygy, are the dualistic component which represents in men the feminine unconscious and in women, the masculine unconscious.[41]
Both the shadow and syzygy can be realized and overcame, however, this task is not an easy one requires the presence of a partner to be recognized.  In the case of the syzygy, this occurs “only through a relation to a partner of the opposite sex, because only in such a relation do their projections become operative.”[42]   It is only after we have overcome these unconscious aspects of ourselves, that we can begin our paths to self-knowledge, which for Jung was “the total personality which, though present, cannot be fully known.”[43]  The realization of this self is the goal of life, because through it we transcend every opposing aspect of our personality so all of our self “is experienced equally.”[44]
Many aspects of these archetypical forms can be seen as occurring in Guido’s memories and dreams, where the archetypes could become activated at a critical moment which the subject “cannot overcome by conscious means.”[45]  His earliest memory is rich in this symbolism, and the magic words “Asa Nisi Masa” are a takeoff on Jung’s concept of the anima.  This, coupled with the focus on matriarchal figures in Guido’s life, can bring forth the conclusion that much of his own anima is taking shape in this sequence.  However, Guido’s own shadow side is also beginning to form here, and can be seen engulfing young Guido at the beginning of the scene.  This shadow side is something Guido would come to hide well in his dealings with others through his persona.  However, Guido could not suppress his shadow forever, and in the harem fantasy when he is forced to resort to violence to control the women, Guido becomes awakened to his own shadow side.  This begins a humbling process in Guido, and leads him to the modesty he is lacking to achieve self-realization and acknowledge his own imperfections.[46]
This process of self-realization continues, as an even deeper symbolism of the anima manifests itself in Claudia, the woman set to play Guido’s ideal woman in white.  When she criticizes Guido’s hero as unsympathetic and claims this character is afraid to love anyone, he finally listens and becomes self aware enough to take positive steps toward coming to terms with his own feminine nature.  As Jung states, overcoming the syzygy, like the shadow, is not easy and often meets with considerable resistance, but they are steps Guido has to take in order to achieve self-realization and solve the crisis that is plaguing him.[47]  Because Guido’s hero is merely a projection of himself, by acknowledging these shortcomings, for the first time in the film he has assessed himself and in the process “succeeded in deeming himself worthy of serious attention and sympathetic interest.”[48]
Guido is next taken to a press conference where the production of his new film is set to be announced, and with both a newfound modestly courage, he finally reveals to the producers that there is, in fact, no film.  Nevertheless, all hope is not lost for Guido, because he is now equipped with this knowledge and acceptance of both his dark side and unconscious feminine nature.  He is quickly reassured he made the correct decision by Daumier, and after a deeply introspective moment of clarity, Guido appears, at least in fantasy, to have achieved this self-realization, and is able to direct all the characters who have appeared in the film together into a circle, where together they will dance in unison.
The circle has a very important meaning for Jung, and of all the geometric forms, he devotes most of his time to discussion of this shape.  Jung felt it corresponded “not to the ego but the self as the summation of the total personality” and signified “nothing less than a psychic centre of the personality not to be identified with the ego.”[49]  In connection with this, Jung was similarly fascinated with the mandala, “a circle used in…yoga as…an aid to contemplation.”[50]  He felt “the true mandala is always an inner image…gradually built up…through imagination.”[51]  It is no surprise then why these circular images were connected with the realization of the self.  However, Jung acknowledges this realization is a most difficult task and “an awareness of the two sides of man’s personality is essential, of their respective aims and origins.  These two aspects must never be separated through arrogance or cowardice.”[52]  Jung felt the realization connected with the symbolism of the circle was something that grew in the individual over time, and as Stubbs illustrates, it had been growing in Guido’s mind since childhood, and developed into the imperfect circle of characters in the harem fantasy, and ultimately the completed circle of characters in the final scene.[53]

While Guido appears to have realized the creative vision he needed to conquer his crisis, because the final sequence is a fantasy, much of what will actually happen to Guido is left up to the mind of the viewer.  There are still many questions regarding how Guido will behave in relation to the characters that populate the circle, and as Stubbs acknowledges, we never are told how this sudden revelation will affect Guido’s relationship with the other characters.  This is seen particularly in the interactions with Luisa, who we had last had seen in a heated argument with Guido.  Now, in his fantasy, she appears to have forgiven him, but we must remember, this is a fantasy and all his marital problems are not solved, an aspect that has caused a variance in interpretation.[54]  The fact that Fellini left Louisa’s decision up in the air should not be a surprise when one considers Jung observation that “practically all the trump cards are in the hands of our opponents.”[55]


[1] . Dir. Federico Fellini. Perf. Marcello Mastroianni and Claudia Cardinale. Corinth Films, 1963-1992.  All references to the film from this point on will be to this version.
[2] Stubbs, John C. “8½.” Journal of Aesthetic Education 9.2 (1975). 106.
[3] Stubbs, “8½,” 106.
[4] Fredericksen, Don. “Two Aspects of a Jungian Perspective Upon Film: Jung and Freud; The Psychology of Types.” Journal of the University Film Assoication 32 (1980). 49.
[5] Fredericksen, 53.
[6] Jung, Carl G. “Psychological Types.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 180-182.
[7] Jung, “Psychological Types,” 180; 229.
[8] Jung, “Psychological Types,” 192-229. 237-266.
[9] Jung, “Psychological Types,” 266-267.
[10] Jung, Carl G. “Individual Dream Symbolism in Relation to Alchemy.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 367.
[11] Jung, Carl G. “Psychology and Literature.” Modern Man in Search of Soul. Trans. Dell, W.S. and Cary F. Baynes. New York: Harcourt Brace Jovanich, 1955. 509.
[12] Jung, “Psychology and Literature,” 509.
[13] Jung, “Psychology and Literature,” 510.
[14] Jung, “Psychology and Literature,” 511.
[15] Jung, “Psychology and Literature,” 509.
[16] Jung, “Psychology and Literature,” 512.
[17] Jung, “Psychology and Literature,” 513.
[18] Jung, “Psychology and Literature,” 513.
[19] Jung, “Psychology and Literature,” 509.
[20] Jung, “Psychology and Literature,” 509.
[21] Goethe, Johann Wolfgang von. Faust. Trans. Walter Kaufmann. New York: Anchor Books, 1963. 63-421.
[22] Jung, “Psychology and Literature,” 509.
[23] Goethe, Johann Wolfgang von. Faust. Trans. Walter Kaufmann. New York: Anchor Books, 1963. 422-503.
[24] Jung, Carl G. “On the Relation of Analytic Psychology to Poetry.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 302.
[25] Jung, “On the Relation,” 311.
[26] Jung, “On the Relation,” 314.
[27] Jung, “Psychology and Literature,” 520.
[28] Stubbs, John C. “Fellini’s Portrait of the Artist as Creative Problem Solver.” Cinema Journal 41.4 (2002), 126; 129.
[29] Bachmann, Gideon. “A Guest in My Own Dreams: An Interview with Federico Fellini.” Film Quarterly 47.3 (1994), 9.
[30] Bachmann, 4.
[31] Bachmann, 13.
[32] Jung, “Psychology and Literature,” 510.
[33] Jung, “On the Relation,” 303-308.
[34] Jung, Carl G. “The Concept of the Collective Unconscious.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 60.
[35] Jung, “The Concept of,” 60.
[36] Jung, “On the Relation,” 321.
[37] Jung, “On the Relation,” 321.
[38] Jung, “The Concept of,” 60-64.
[39] Jung, Carl G. “Aion: Phenomenology of the Self.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 145.
[40] Boeree, C. George. “Archetypes.” Boeree, 2. Introduction to C.G.Jung. 10 Dec. 2010. <http://www.studiocleo.com/librarie/jung/boeree3main.html>.
[41] Jung, “Aion,” 148-153.
[42] Jung, “Aion,” 145; 161.
[43] Jung, Carl G. The Undiscovered Self. Trans. R.F.C. Hull. New York: The New American Library, 1964. 61.  Jung, “Aion.” 142.
[44] Boeree, C. George. “The Self.” Introduction to C.G.Jung. 10 Dec. 2010. <http://www.studiocleo.com/librarie/jung/boeree5main.html>.
[45] Jung, The Undiscovered Self, 80.
[46] Jung, The Undiscovered Self, 116.
[47] Jung, “Aion,” 145.
[48] Jung, The Undiscovered Self, 101.
[49] Jung, “Individual Dream Symbolism,” 362; 367.
[50] Jung, “Individual Dream Symbolism,” 559.
[51] Jung, “Individual Dream Symbolism,” 560.
[52] Jung, “Individual Dream Symbolism,” 372.
[53] Jung, “Individual Dream Symbolism,” 364. Stubbs Portrait, 126.
[54] Stubbs, “Fellini’s Portrait,” 129.
[55] Jung, The Undiscovered Self, 75.

Bibliography
1.      . Dir. Federico Fellini. Perf. Marcello Mastroianni and Claudia Cardinale. Corinth Films, 1963-1992.
2.      Bachmann, Gideon. “A Guest in My Own Dreams: An Interview with Federico Fellini.” Film Quarterly 47.3 (1994). 2-15.
3.      Boeree, C. George. “Archetypes.” Boeree, 2. Introduction to C.G.Jung. 10 Dec. 2010. <http://www.studiocleo.com/librarie/jung/boeree3main.html>.
4.      Boeree, C. George. “The Self.” Introduction to C.G.Jung. 10 Dec. 2010. <http://www.studiocleo.com/librarie/jung/boeree5main.html>.
5.      Goethe, Johann Wolfgang von. Faust. Trans. Walter Kaufmann. New York: Anchor Books, 1963. 503.
6.      Fredericksen, Don. “Two Aspects of a Jungian Perspective Upon Film: Jung and Freud; The Psychology of Types.” Journal of the University Film Assoication 32 (1980). 49-57.
7.      Jung, Carl G. “Aion: Phenomenology of the Self.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 145.
8.      Jung, Carl G. “Individual Dream Symbolism in Relation to Alchemy.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976.
9.      Jung, Carl G. “On the Relation of Analytic Psychology to Poetry.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 301-322.
10.  Jung, Carl G. “Psychology and Literature.” Modern Man in Searc h of Soul. Trans. Dell, W.S. and Cary F. Baynes. New York: Harcourt Brace Jovanich, 1955. 507-520.
11.  Jung, Carl G. “Psychological Types.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 178-269.
12.  Jung, Carl G. “The Concept of the Collective Unconscious.” The Portable Jung. Ed. Joseph Campbell. Trans R.F.C. Hull. New York: Penguin, 1976. 59-69.
13.  Jung, Carl G. The Undiscovered Self. Trans. R.F.C. Hull. New York: The New American Library, 1964. 125.
14.  Stubbs, John C. “8½.” Journal of Aesthetic Education 9.2 (1975). 96-108.
15.  Stubbs, John C. “Fellini’s Portrait of the Artist as Creative Problem Solver.” Cinema Journal 41.4 (2002). 116-131. .

Friday, March 9, 2012

The Influence of Wittgenstein’s ‘Middle-Period’ Philosophy of Mathematics on his Later Account of Possibility.
Michael Nielson

Ludwig Wittgenstein held various beliefs through the publication of the Tractatus Logico-Philosophicus that he would later come to repudiate in the Philosophical Investigations.  One of these was the atomistic notion that sentences which could be understood could also be analyzed into elementary propositions, termed by Diane Gottlieb in “The Tractatus View of Rules” as ‘the truth-functional thesis’.  One of the primary reasons for this rejection of the truth-functional thesis in the Tractatus Logico-Philosophicus is its conception of rules being motivated by an idealized picture of language’s use.    In the next phase of his career, he would become preoccupied with the philosophy of mathematics and forge an attack on the idea of infinite mathematical sets as anything more than possibilities due to their reliance on finite mathematical calculi.  While in the TLP rules were given a much more strict application and “alone determined meaning”, in the PI meaning and truth could only “be accounted for only in the context of practice”[1]. During this later period, he would return his focus to language, and though it would deviate from the middle period, his discoveries about the nature of mathematics would come to transform many of Wittgenstein’s ideas through the development of language-games, allowing him to account for the variation and possibility in our languages. 
            In order to illustrate the effect his philosophy of mathematics had on the evolution of his conception of rules, I will first outline the view of rules in Wittgenstein’s Tractatus, and examine both syntactical and translation rules.  Next I will detail how his preoccupation with mathematics led him to classify various mathematical propositions, such as set theory and infinite numbers, not as mathematical calculi in and of themselves, but instead as a mere infinite possibilities.  Finally, I will explain how, despite the differences between the two, this questioning of mathematical propositions would play a factor in the Philosophical Investigations repudiation of the truth-functional thesis within logical atomism, as well as the rejection of a rule functioning inside any sort of mental mechanism.

The Early Wittgenstein  
In the only work published during his lifetime, the Tractatus Logico –Philosophicus(TLP), Ludwig Wittgenstein expounded a picture-theory of meaning.  At the foundation of this picture-theory was logical atomism.  According to this doctrine, both meaning and language were examined “in terms of logical form and definiteness of sense”[2].   This theory held the essential metaphysical form of the world and the logical form of language were isomorphic with one another due to their similar structure[3].   Consequently, the objects that the names stood for had to be indestructible,[4] and therefore definite.
Central to Wittgenstein’s atomism was ‘the truth-functional thesis’, which “held that on any occasion when a sentence of ordinary language was meant and understood, the sentence was analyzable into elementary propositions of which it was a logical product”[5].  Implicit within the truth-functional thesis, according to Diane Gottlieb, are rules[6].  These rules are defined in two distinct ways within the TLP, as rules of translation and rules of syntax.
3.334 – The rules of logical syntax must go without saying, once we know how each individual sign signifies
3.343 – Definitions are rules for translating from one language into another.  Any correct sign-language must be translatable into any other in accordance with such rules: it is this that they all have in common.[7]
The rules of definition “play a role in the processes of meaning and understanding”[8].  The rules of logical syntax on the other hand were much more definite, and though it is unclear how Wittgenstein saw them, Hans-Johann Glock theorizes he felt that they mirrored “the essence of reality”[9].  A single logical syntax was “the logico-metaphysical structure which all meaningful languages…must have in common”[10].  However, during his post-TLP sabbatical from philosophy, the Vienna Circle would develop their own unique interpretation of the rules of syntax as “arbitrary conventions governing the use of signs”[11].  Despite these differences both interpretations share the commonality that the only contexts they were able to be discussed was with regard to their use.
            As opposed to these implicit rules, it was explicit within the truth-functional thesis that “all sentences with sense…are in perfect logical order” and possess “a determinate or definite sense” with little account taken for possibility[12].  It was in reference to this that the rules of the TLP lacked any sense of possibility.  This is one of the multiple reasons I theorize why Wittgenstein would came to question the TLP and view logical atomism as dogmatic in character[13].  These factors caused him to see at the root of logical atomism was “a false and idealized picture of the use of language” that would cause him to first leave philosophy entirely before ultimately reassessing his philosophical system upon his return[14].
The Middle Wittgenstein & The Philosophy of Mathematics
Though his philosophical system would progress and evolve after the completion of the TLP, a formalistic conception of mathematics was always foundational to his system[15].  Formalism in mathematics is identified as “the philosophical theory that formal (logical or mathematical) statements have no meaning but that its symbols (regarded as physical entities) exhibit a form that has useful applications”[16].  These mathematical views would take center stage in the middle period of his career after Wittgenstein attended Vienna lecture entitled ‘Science, Mathematics and Language’ by L.E.J Brouwer on March 10, 1928 and became more aware of the interpretations of the TLP[17].  This period is often seen as a black hole in Wittgenstein’s philosophy and is termed by Steve Gerrard in “Wittgenstein’s Philosophies of Mathematics” as ‘the calculus conception of mathematics’.
During this period Wittgenstein would carry from the TLP “his strong brand of formalism, according to which…we invent mathematics, bit-by-little-bit”[18].  Due to being our own creation mathematical propositions only have “intra-systemic meaning, which is wholly determined by its syntactical relations to other propositions of the calculus”[19].  Mathematics was “a closed, self-contained system” where rules were given a much larger role and “alone determined meaning, and thus became the final and only court of appeal”[20].  This reformulation of mathematical rules was only the first of the changes that will be discussed as ultimately crossing over into his later periods philosophy of language and his conception of rules.
Despite these loyalties to a formalistic idea of mathematics, in this period much of his focus would shift to rejecting “quantification over an infinite mathematical domain”[21].  The idea of an infinite class does not have the same syntax as a finite class, because “an infinite mathematical extension is a contradiction in terms”[22].  To further illuminate this, Wittgenstein is quoted in the Philosophical Grammer(PG) as saying “the infinite is understood rightly when it is understood, not as a quantity, but as an ‘infinite possibility’”[23].  We are only able to know it as a possibility through acquaintance with the “criteria for the truth of similar propositions”[24].  In other words, infinite mathematical sets can only be understood in reference to finite propositions because “there are no criteria for mathematical correctness outside the rules of individual calculi”[25].  We are unequipped to even consider grasping anything more than this mere possibility.
Regardless of this rejection of infinite number sets, Wittgenstein did not completely object to the idea of an underlying mathematical reality[26].  What he rejected was “a conception of mathematical reality that is independent of our practice and language” and “is capable of overruling how we actually do mathematics”[27].  This ‘faulty conception of mathematical reality’ can be seen as leading him to critique various mathematical theories surrounding the concept of an infinite set.  Two examples of these theories examined by Victor Rodych in the Stanford Encyclopedia of Philosophy include ‘Fetmat’s Last Theorem’ and ‘Goldbach’s Conjecture’.  The formal statements for both appear as follows:
                            -----------------------------------------------------------------
----------------------------------------------------------------
-----------------------------------------------------------------
            Goldbach’s conjecture, which to this day has yet to be proven, was first discovered in 1742 in a letter written to Euler by Goldbach, in which he states “at least it seems that every number that is greater than 2 is the sum of three primes[30].  The other example, Fermat’s last theorem, was first discovered posthumously in the margin of his copy of Arithmetica by Diophantus, appeared as follows:
"It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain”[31].
What both of these theories had in common is that the use of the variable ‘n’ designates the nth number, and leads to a set of ‘any number’ or ‘every number’ that is greater than ‘x’.  Because these theories are based around this conception of a set including any or every number greater than ‘x’, “contrary to our proof it could be false that there is not a largest number”[32].  We can never fully prove these equations because when we examine an infinite set, it “resides only in recursive rules”.  Further exploration into Wittgenstein’s reasoning behind rejecting anything more than this possibility when dealing with infinite number sets will further illuminate the evolution of the concept of rules and why at the base of this was a “faulty conception…of a mathematical reality that is capable of overruling how we actually do mathematics”[33]
The Later Wittgenstein and his Return to Language
Around 1944 this faulty conception of mathematics would cause Wittgenstein to once again return his focus to language, in a period viewed by Gerrard as ‘the language-game conception of mathematics’[34].  This both deviates from and carries over many of the concepts within the calculus conception[35].  Although he continues his “formalistic conception of mathematical propositions and terms” he would reject the calculus conception based on mistakes that “come from reaching too far and from unnecessarily restricting the scope of language”[36].  These mistakes involving the restricted scope of language run parallel to many of the same errors that were fundamental to the critique of the truth-functional thesis and rules of syntax, in that both failed to take full account for possibility.
This new conception of both language and mathematics would strip itself of his previous mistakes regarding definiteness by taking “account for the change and growth of mathematics”[37] and both meaning and truth were “accounted for only in the context of a practice”[38].  This would move the idea of a criterion for mathematical correctness from the rules of the calculi to the mathematical practice itself[39].  The reasoning behind this shift from the calculus conception can be seen in its inability to account for change, growth and possibility in mathematics and language[40].  This section of the paper will introduce just a few aspects of the PI and detail how they both deviated from and built upon his earlier career, leading him to take account for this possibility in language by viewing it within “the context of a practice”[41].
One of these aspects was another dualism within Wittgenstein’s philosophy identified by Gottlieb, only this time instead of identification with rules, it concerned connections.  The first of these connections are set up by a calculus, here the result “is not dependent upon, nor vulnerable to, the accidents of fortune”[42].  The other is set up in a mechanism where “the effect can unexpectedly be determined by some unforeseen occurrence such as the malfunctioning of the mechanism for various reasons”[43].  Through the acknowledgement of this mechanistic aspect of connections two key notions in the PI are addressed.  The first is that the “rules of language ought not be presumed to have strict application” and the second is the logical inconsistency in thinking of a rule of language “as something functioning in a mental medium”[44].
Sections 193 and 194 of the PI concern themselves with this distinction between both types of connections by using an example of machines to both expound his new view and further the critique of syntactical and translational rules[45].  The distinction he has made between the ideal machine, or machine-as-symbol, and the actual machine builds upon the notion of the rules themselves functioning outside of the mental mechanism.  The ideal machine can be seen as merely a blueprint or design of the machine, it does not account for any sort of possibility that may occur and acts “as if the parts of the machine ‘could do no other’”[46].  This classification of an ideal machine can be seen as acknowledging many of the same problems he had found within the conception of anything more than a possibility of an infinite set. 
In PI 193 Wittgenstein draws a parallel between the assumption that someone “will derive the movement of the parts from” this ideal machine and telling someone “it is the twenty-fifth number in the series 1, 4, 9, 16, …”[47].  This comparison serves as an example that similar to middle period’s criticisms of Goldbach’s Conjecture and Fermat’s Last Theorem, merely telling someone the twenty-fifth number in a sequence is unable to prove anything more than a possibility.  The most we can comprehend is a possibility, “the movement of the machine-as-symbol is predetermined in a different sense from that in which the movement of any given, actual machine is predetermined”[48].
On the other hand, the actual machine does take account for actual movements, and the possibility of the machine bending, breaking or evolving[49], placing him at odds with the view presented in the TLP[50].  These possibilities of movement are identified by Wittgenstein through a vague example as “the shadow of the movement”[51]  However, Wittgenstein takes this notion of possibility to be much more foundational, and critiquing the picture theory, he states that a possibility of movement “stands in a unique relation to the movement; closer than that of a picture to its subject”[52].  Because of this possibility “we are at liberty to construct novel calculi, constrained only by the demand for consistence and considerations like ease of explanation and avoidance of puzzlement”[53].  This possibility within language depicted by the machine analogy can be seen as building upon many notions of his philosophy of mathematics.  The analogy serves as an example of the influence Wittgenstein’s belief that we were only able know an infinite set as a possibility, as opposed to being the genuine result of a mathematical calculi, had on both his philosophy of language during the final stages of his career and criticisms of both his conception of rules and the truth-functional thesis within the TLP.
Conclusion
            Wittgenstein was able to take account for the possibility within both language and mathematics through the more fluid and constructivist approach developed in the PI.  This stands in a sharp contrast to the more definite view found in the TLP.  Many of these contrasts between the early and late system were greatly influenced by the function rules were given to perform.  Wittgenstein’s mathematical formalism and criticisms of theories regarding infinite sets shed light motives behind this shift in thought, of which the most notable consequences were the more fluid account of language.


Notes


[1] Gerrard, Steve. “Wittgenstein’s Philosophies of Mathematics.” Synthese 87.1 (1991): 126
[2] Gerrard, 131
[3] Glock, Hans-Johann. “The Development of Wittgenstein’s Philosophy.” Wittgenstein: A Critical Reader. Ed. Hans-Johann Glock. Massachusetts and Oxford: Blackwell Publishing, 2001:7.
[4] Glock. “The Development of Wittgenstein”, 8.
[5] Gottlieb, Diane F. “Wittgenstein’s Critique of the ‘Tractatus’ View of Rules.” Synthese 56.2 (1983):  239.
[6] Gottlieb, 241.
[7] Wittgenstein, Ludwig. Tractatus Logico-Philosophicus. Trans. D.F. Pears and B.F. McGuinness. London and New York: Routledge Classics, 2008: 20-21.
[8] Gottlieb, 241.
[9] Glock. “The Development of Wittgenstein”, 11.
[10]Glock, Hans-Johann. “The Development of Analytic Philosophy: Wittgenstein and After.”  The Routledge Companion to Twentieth Century Philosophy. Ed. Dermot Moran. London and New York: Routledge, 2008: 82.
[11] Glock. “The Development of Wittgenstein”, 11.
[12] Gottlieb, 240-241.
[13]Biletzki, Anat. “Ludwig Wittgenstein.” Stanford Encyclopedia of Philosophy. 23 Feb. 2007. http://plato.stanford.edu/entries/wittgenstein, 29 Nov. 2009. 3.1.
[14] Gottlieb, 242.
[15] Rodych, Victor. “Wittgenstein’s Philosophy of Mathematics.” Stanford Encyclopedia of Philosophy. 23 Feb. 2007. http://plato.stanford.edu/entries/wittgenstein-mathematics, 16 Dec. 2009. 1.0.
[16] “Formalism.” Princeton WordNet. Princeton University, http://wordnetweb.princeton.edu/perl/webwn?s=formalism. 8 May. 2010.
[17] Rodych, 2.0.
[18] Rodych, 2.1.
[19] Rodych, 2.1.
[20] Gerrard, 126.
[21] Rodych, 2.2.
[22] Rodych, 2.2.
[23] Rodych, 2.2.
[24] Rodych, 2.4.
[25] Gerrard, 131.
[26] Gerrard, 128.
[27] Gerrard, 128.
[28] “Fermat’s Last Theorem.” Wolfram Alpha. http://www.wolframalpha.com/input/?i=fermat%27s+last+theorum. 8 May 2010.
[29] “Goldbach Conjecture.” Wolfram Alpha. http://www.wolframalpha.com/input/?i=goldbach+conjecture. 8 May 2010.
[30] Weisstein, Eric W. "Goldbach Conjecture." Wolfram MathWorld. http://mathworld.wolfram.com/GoldbachConjecture.html. 8 May 2010.
[31] Weisstein, Eric W. "Fermat's Last Theorem." Wolfram MathWorld. http://mathworld.wolfram.com/FermatsLastTheorem.html. 8 May 2010.
[32] Gerrard, 130.
[33] Gerrard, 128.
[34] Gerrard, 126.
[35] Gerrard, 126.
[36] Gerrard, 132.
[37] Gerrard, 132.
[38] Gerrard, 126.
[39] Gerrard, 131.
[40] Gerrard, 132.
[41] Gerrard, 126.
[42] Gottlieb, 243.
[43] Gottlieb, 243.
[44] Gottlieb, 243.
[45]Wittgenstein, Ludwig. Philosophical Investigations. Trans. G.E.M. Anscombe. Massachusetts and Oxford: Blackwell Publishing, 1999. 77-79.
[46] Gottlieb, 247.
[47] Wittgenstein. Philosophical Investigations, 79.
[48] Wittgenstein. Philosophical Investigations, 77-78.
[49] Wittgenstein. Philosophical Investigations, 79
[50] Glock. “The Development of Analytic”, 82.
[51] Wittgenstein. Philosophical Investigations, 79.
[52] Wittgenstein. Philosophical Investigations, 79.
[53] Glock. “The Development of Analytic”, 82.