Beauvoir, Irigaray, and Philosophy
Beauvoir, Irigaray, and Philosophy
Abstract and Keywords
Luce Irigaray’s view of her relationship to Beauvoir’s work is that “there are important differences between our positions.” This should not be surprising given that these two philosophers belong to different even if overlapping philosophical eras. Beauvoir is identified primarily with phenomenological–existentialism and Irigaray with psychoanalysis and linguistics. This essay takes up those differences from an ontological and epistemological point of view suggested by a number of feminist philosophers but not fully examined in the work of Beauvoir and Irigaray. This includes Beauvoir’s rejection of dualist thinking produced by the binary logic of the Law of Excluded Middle, and Irigaray’s critique of formal logic based on her psychoanalytic perspective. Beginning with Beauvoir and moving from there to Irigaray, the essay takes up the question of the ontological and epistemological structures utilized by each of these two feminist philosophers with an eye to their subsequent ethical implications.
Exploring the distance between the philosophy of Simone de Beauvoir and that of Luce Irigaray may be mediated initially by Irigaray’s comments on her relationship to Beauvoir’s work. In her own words, “there are important differences between our positions.”1 This should not be surprising, given that these two philosophers belong to different, even if overlapping, philosophical eras, Beauvoir being identified primarily with phenomenological–existentialism and Irigaray with psychoanalysis and linguistics. How any contemporary philosopher approaches those differences says as much about her own positions and interests as those of Beauvoir and Irigaray. This essay takes up those differences from an ontological and epistemological point of view suggested by a number of feminist philosophers but not fully examined in the work of Beauvoir and Irigaray. Tove Petterson has designated Simone de Beauvoir’s ontological position an effect of her reservations concerning that of Jean-Paul Sartre. These reservations, she argues, produced a relational ontology commensurate with Beauvoir’s ethics. Her claim is that Beauvoir charts a middle way that navigates between individualism and collectivism, (p.217) idealism and materialism, mind and body.2 Pettersen further articulates this epistemologically in terms of Beauvoir’s clear rejection of dualist thinking produced by the binary logic of the Law of Excluded Middle. Even so, Val Plumwood finds that Beauvoir did not develop her criticism of dualistic logic with adequate precision and so did not escape the criticisms later leveled at her by other feminists, including Irigaray.3
Given these precedents, it is my initial task to show that all of Beauvoir’s work, but especially The Second Sex, seeks to undo the binary structure enforced by the logic of the Law of Excluded Middle and thereby makes possible a logic and ethics of ambiguity. This brings us to the further question of how Beauvoir’s efforts might compare with logical and ontological structures and their ethical consequences as employed by Irigaray. In her account of Irigaray’s critique of formal logic, Marjorie Hass points out that, while logicians focus only on the formal properties of logical concepts, Irigaray’s psychoanalytic perspective includes the Imaginary meaning of logical connectives. So any discussion of Irigaray’s positions regarding logic and science will have to take into account her view of the science of psychoanalysis. Beginning with Beauvoir and moving from there to Irigaray, let us take up the question of the ontological and epistemological structures utilized by each of these two feminist philosophers with an eye to their subsequent ethical implications.
8.2. The Binary Logic of the Excluded Middle
Undoing the binary structure of the excluded middle does not imply that it has no use in any context. Rather, it implies that Beauvoir is questioning its value for human life and action and that she does this in lieu of the logic of Jean-Paul Sartre’s ontology in Being and (p.218) Nothingness. The Principle of Excluded Middle states that for a given proposition “P,” either “P” or “not-P” must be true and they cannot both be true; thus it affirms the fundamental principle of binary thinking and the logic of identity.4 This means that being cannot be both some x and its negation; only one can be true and at least some position x is true unless superseded by its negation. This is the logic that Beauvoir will counter with her own logic of ambiguity.
Sartre argues that we humans are encompassed by nothingness, the permanent possibility of nonbeing, such that what being will be arises on the basis of what it is not. By negating the being that we have been, we declare the past to be false and the present to be true. Logically, we may formulate this by saying that “being is x and outside of x it is nothing,” yet Sartre also wants to maintain that nonbeing is real.5 He argues that the ground of any perception is an original nihilation necessary for a figure to appear but into which it melts like unwanted faces in a cafe when one is searching for someone who is not there. The one who is not there thus appears as a nothingness on a ground of nihilation, a nothingness that is real, not merely thought. In this model, negation is discrete, that is, it is an abrupt break in continuity; an original, irreducible event, but also a perpetual presence in and outside us.6
Within the logical order, the asymmetry of true or positive propositions is symbolized by the representation of negation through the addition of a negation symbol to the positively expressed proposition “P,” thus “P” and “¬P.” Although for some logicians, such as Göttlob Frege, “P” and “¬P” coemerge as logically equivalent, for others, such as Ludwig Wittgenstein, “the sense of ‘¬P’ cannot be understood unless one understands the sense of ‘P.’ ”7 In other words, when the negative is thought through strictly, it cannot appear as an object of thought. Wittgenstein took this to the point of declaring that, outside of symbolic language, nothing can even be said. Sartre understands negation in the first manner, as coemergent with being, (p.219) even as negation reveals being through objectification, denies its transcendence, and reduces it to mere facticity. For this reason, some philosophers interpret negation as constituting a structural relation of dominance and erasure because it constructs difference or otherness in terms of exclusion or by denial of transcendence.8
For Sartre, by denying that I am this or that particular being, I can make the world come into being. Sartre maintains that things in the world reveal their resistance and adversity in relation to our own “instrumental complex,” our plans and actions over against which objects are too heavy, too big, too fragile, or too threatening. Our body is inapprehensibly given, even though it is the necessary condition of all actions. In our actions, we live our body through tools that are not our body, but that we utilize and that first indicate our body to us. The body is therefore a being-in-the-midst-of-the-world, an instrument, and our body is for us always already surpassed toward instrument–objects in the world, and is itself an obstacle to surpass in order to be in the world.9
The binary relationship with our own body is repeated in contact with the Other. For Sartre, when the Other looks at me, she possesses me. This possession is her consciousness of possession, proof that she has a consciousness and proof of her positive existence ‘P.’ In this moment, I recognize my object state and I also have proof that the Other is a consciousness. Negating my transcendent being, the other causes me to be a being at all. I am responsible for this being but I am not its foundation, as it is contingently given to me through the other’s negations of me (¬P). However, my project of recovering myself necessarily involves negation and assimilation of the other, possession of the other, making the other into an object. So we may question, on logical grounds, Sartre’s claim that we begin with the idea that every question opens the possibility of a negative reply.10 Sartre, it appears, is caught up in the binary logic of the excluded middle that operates point by point, moment by moment, so that being (p.220) and nonbeing are arranged like atoms in space, and, according to his own formula, each one nihilates the other, because, given “P” and “¬P,” one must be true and only one can be true.
8.3. Caught in the Excluded Middle
This logic is exemplified for Beauvoir in The Second Sex by the phrase “he is male,” which is said to be a source of pride, whereas “she is female” is unequivocally derogatory. She finds this pattern repeated throughout our interpretations of the data of biology and throughout the life of women. Yet, as Beauvoir pointedly argues, biologically, male and female are merely two types within a species differentiated according to their reproductive function, which is in many species not at all clear-cut.11 For single-celled creatures, each cell divides and subdivides. For more complex, many-celled creatures (metazoans), reproduction may be asexual by fission or by blastogenesis, new buds separating into new individuals, or by parthenogenesis, without any male contribution.12 That there are two types of gametes means little, and all we can say for sure is that several kinds of reproduction occur in nature.13
Beauvoir argues valiantly (although I think incorrectly) that Plato, Aristotle, St. Thomas, and even Hegel do not insist on anything but the contingent nature of all sexuality. Nevertheless, Beauvoir concedes that, up until at least 1879, when the mammalian egg was “discovered,” philosophers and scientists maintained that the male was superior, the source of activity, and the female passive and lacking, used up, nihilated by the newly formed being. But in nature, she maintains, there is no a priori binary structure and no hierarchy; egg and sperm each contain a single set of equivalent chromosomes, father and mother play an equal role in heredity, and both are suppressed in the formation of a new whole.14
(p.221) Thus, she concludes, it is foolhardy to deduce from reproduction that the female human does not actively transcend. Yet women struggle against domination by their own species, a role that biology does not condemn them to.15 If biology does not condemn women to the Law of Excluded Middle, then what does? It appears that the system of binaries is largely an effect of social norms. As a child, the female “learns” that the lack of a male organ is shameful and a deprivation, that menstruation is dirty, and that her interest in sexuality is impure.16 The adolescent female is embarrassed by the changes her body undergoes. In addition, her mother imposes family tasks and errands, housework after school, and she is not free to roam with her friends.17 She is expected to behave like a “well-bred young girl,” and any aggressive behavior dooms her as she struggles to establish the truth of her own inferiority, ultimately making herself into an adorable object.18 Caught by the Law of Excluded Middle, she is dazzled but frightened by males, and she negates the fright through the adoration of a distant hero, a movie star or rock star whom she will never encounter.19 Her only experience as a free being is in nature :
She finds in the secret places of the forest a reflection of the solitude of her soul and in the wide horizons of the plains a tangible image of her transcendence . . . . In the rush of water, the shimmer of light, she feels a presentiment of the joys, the tears, the ecstasies, she has not yet known; the ripples of the pool, the dappled sunlight, give vague promise of these adventurings of her own heart.20
Only in nature does the girl have the opportunity to live the logic of ambiguity, existing as both organism and consciousness, each without negating the other.
How this contrasts with her experience of sexual initiation when she becomes physically and mentally divided against herself! Her (p.222) sexual pleasure is obtained in opposition to the spontaneous surge of her sensuality.21 The positive phenomena that arouse her, the expenditure of vital energy, are countered by prohibitions, prejudices, and exactions to which she must submit.22 Whether the man takes or gives pleasure, the asymmetry is there; she is the instrument through which he is his own body. In spite of this, insofar as the woman’s entire body is moved by desire and sexual excitement, the woman retains her subjectivity, not through negation, but through union with the partner when both give and both receive, an ambiguous situation indeed.23
Asymmetry and opposition can be forestalled in a relationship between two women in which the desire to possess and to give the other all coexist and each is ambiguously and simultaneously subject and object.24 For Beauvoir, lesbian sexuality, its manifold combinations, transpositions, and exchanges, affirms that there is no single determining factor in sexual matters and that each woman’s erotic preferences are an expression of her general outlook on life.25 Only the social realm enforces asymmetry and the excluded middle. Ideally, at least, sexual relations between women remain outside of institutional sanctions and mores as well as social conventions, so their respective roles are balanced by their psychological tendencies in accordance with their total situation.26 They are situated, but to a much lesser degree.
So we can see that since, for Beauvoir, marriage is offered to women by society, this does not bode well for their determination by the Law of Excluded Middle. Economic independence can upset the asymmetry, making marriage a reciprocal union of two independent persons; nevertheless, social conformity is difficult to evade. Arranged marriages are still common in many cultures, and adolescent girls are forced into marriage with old men, ensuring the requisite demand for virginity. Just as often, the woman is confined to the home. But Beauvoir does not actually contrast the home to the (p.223) workplace; instead, it stands in contrast to nature: “When she was a girl, the whole countryside was her homeland, the forest was hers.”27 With marriage, the vast expanses of nature’s variability collapse into the confines of the house and the repetitive tasks of housekeeping, which deny the woman the experience of choice and the experience of herself as a being without negation.
Lacking the regular encounter with choice and ambiguity, confined more and more to the role of not being an agent, not choosing, not expressing her psychological or sexual tendencies, the woman becomes—but she becomes the “¬ P” of the excluded middle. For a woman, marriage and home make use of and demand this structure, but it may be understood ontologically as her existential estrangement from the relationship to nature and subsequently from the practice of making choices. Estrangement from nature’s ontological ambiguity makes of marriage a series of fixed points at which each moment of the woman’s existence, her connection to her own past and to an open future, are immediately negated.
Meanwhile, a man also defies nature’s ambiguity and practices the techniques of reasoning. The affirmation of contradiction and the commitment to asserting the truth of some proposition “P” and thus the falseness of “¬P” come easily to him. But his reasoning is put to use as a form of violence and domination.28 As an actor in the world, he now knows what is true and what is not true. The woman struggles with this, feeling truth lies elsewhere than in this form of opposition, but lacking confidence and lacking any significant endorsement from society for her reasoning, her faith in the reality of the logic of ambiguity is easily undermined. Even if her version of truth is that love is an impulse toward a person whose existence is separate and distinct from her own and cannot be negated by her own consciousness of existence, choices, and freedom, she receives no support for this in her life.29
Instead, she is turned toward maternity as one more instantiation of negation. Much of Beauvoir’s chapter on the mother deals with (p.224) social and legal restrictions on abortion and contraception as well as the psychological and physical dangers of pregnancy and miscarriage for the woman. Contraception and legal abortion allow women to undertake motherhood in freedom, yet many women throughout the world are denied this choice.30 And even if pregnancy is freely chosen by the woman acting as a conscious and free human being, she is negated by those members of society who view her swollen body with fear or contempt.31 Because babies or young children are unable to give value to the relationship to their mother, the woman remains alone. She must justify her motherhood for herself; yet this does not make her a saint: “Maternity is usually a strange mixture of narcissism, altruism, daydreaming, sincerity, bad faith, devotion, and cynicism.”32 The situation is worsened if the woman feels herself to be inferior, lacking any independent grasp on the world or on her future. Whether her behavior toward the child is capricious, domineering, masochistically devoted, demanding, superior, persecutory, oppressive, or calm and reasonably happy depends on the psychological, moral, and material situation the mother finds herself in. A woman who is well balanced and aware of her responsibilities and options is able to be a good mother, but too often women hope to find in motherhood the warmth and love missing in the rest of their lives.33
All of these life events take place, not in nature but in society, which adheres to certain rational principles that govern behavior and regulates the woman’s appearance. The single girl must attract prospective husbands by exhibiting herself, wearing bright colors or lacy dresses, but the mature woman must not do this; she must modestly cover up. Each moment in life has its atomized determinations that the next step negates. Even women who scorn conventions dress in conformity with their status, and, once dressed, they must preserve themselves from good food that is fattening, good wine that ruins the skin and health, smiling or laughing that brings wrinkles, sun that damages the skin, rest that means missing late-night fun, work that (p.225) makes one weary, love that tires, embraces that mess up the hair and clothes, maternity that disfigures.34 Parties and receptions, if they are a pure generosity, a service rendered to others, are truly celebrations, but as with dress, social convention quickly atomizes them. Each dinner must be negated by an even better return invitation, each party negated by an even better party, and this occurs always in relation to that other Other. Women do not form an independent society, but are a subordinate part of the larger society governed by males, organized by males so that each woman must face the masculine world alone.35 Their talents, intelligence, and appearance are subject to negation coming from the society that includes other women as well as husband, father, brother, or lover.36
In spite of recent gains, life outside of marriage can be harsh, particularly for women without education or wealth. Women still turn to prostitution or they are forced into it by pimps, boyfriends, husbands, even parents. Well integrated into a society that needs their services and cognizant of their status as objectified, prostitutes range widely from children to old women, drug addicts to call girls. Although they are expected to be servile, to serve as the instrument so that their patrons may be their body, insofar as they know this, conflicts between their role and their reality are constant. Subject to everything from fear and poverty to extreme boredom and confinement, their career proceeds under the watchful gaze of the male who makes them exist through their objectification, their negation.37
Yet, the Law of Excluded Middle is not gendered masculine. It exists for anyone to use, but generally in their lives and in their daily tasks, women do not find it useful because it is inadequate to their reality.38 Women tend to preserve and conserve rather than destroy, negate, and begin from nothing.39 When they do protest, seeking to expose the faults of the male use of binary thinking, their tears and violent scenes remain a testament to their subordination to men, ending sometimes in suicide or madness.40 In these situations, (p.226) women are told some facts; Then they are told, “either you agree or you do not agree,” the middle is excluded, and they are accused of being obstinate and illogical if they do not consent.41 According to Beauvoir, women recognize that there is no fixed truth of this nature. Where men see facts, binaries, and identity, atomistically true or false propositions, and the weird version of causality that is the negation of the past moment by the present moment, women remain suspicious. All values seem ambiguous and universal principles of morality extolled by men a hoax.42
Men among men proclaim the truth of the Law of Excluded Middle, but privately, in order not to destroy one another, to maintain social relations (even when they despise one another), they utilize a universal preventative moral code. They know that in a male-organized society, at any moment another man can exercise his freedom, transcend his current situation and negate the past by objectifying another man. As this does sometimes happen, they mostly cautiously distance themselves from one another. Since it is safer not to fully exercise the logic of the excluded middle in the company of other men, they save it, it seems, for women, without the accompanying moral distance. Women are there to be negated, not to transcend, and toward them men can be tyrannical, sadistic, and violent, or puerile, masochistic, and querulous.43
Respected gentlemen take up with prostitutes; intellectuals proclaim their disinterest in money and possessions but nevertheless exploit others to attain them; partners and husbands demand, whether in housekeeping or sex, that women make themselves an object for them, yet that they do it of their own free will, feigning independence at the moment they are most obedient. Once again, women are told some facts, and then, everywhere and all the time, they are told “either you agree or you do not agree.”44 Failure to agree may result in insults, severe treatment, hostility, deprivation, or aggression.
(p.227) So it seems that, for Beauvoir, only in relation to nature, are women free. Women exist in a masculine universe, but let us realize, she says, that even as male society enslaves nature, nature still dominates that society. “Woman is justified by this equivocation in finding more verity [more truth!] in a garden than in a city, in a malady than in an idea, in a birth than in a revolution.”45 But how to do this? Women, like men, are an existence with transcendence. This means that women give value to a domain by transfiguring it.46 The possibility of transfiguring existence can be both thought and realized, but not through Stoicism, something Beauvoir suggests in The Second Sex but rejects in The Ethics of Ambiguity. If women are to reject the deception, the negation demanded by men, there must be a logic that gives them the audacity to formulate an alternative social order and ethics.
The title of Beauvoir’s treatise on ethics, The Ethics of Ambiguity, provides an invitation and incentive to think past the binaries of the excluded middle. Minimally, by name alone, Beauvoir can be said to put into play a logic of ambiguity, one that makes possible an ethics of ambiguity. It is here that we may correlate Beauvoir’s conception of ambiguity with certain aspects of the logic called Intuitionism.
Philosophically, Intuitionism can be traced back to Descartes, who claimed that knowing requires acts of immediate mental apprehension, intuitions, that are the sole source of true and new knowledge.47 Intuitionists also claim Henri Bergson for his distinction between duration and Newtonian moment-by-moment notions of time, such as that utilized by Sartre. In contrast to the atomistic conception of the moment of choice and its subsequent negation, Bergson describes duration as the form that the succession of our (p.228) conscious states assumes when our ego lets itself live—not separating its present from its former states—as when we perceive the notes of a melody floating, each one, in and through the other.48 Intuitionism was brought to full flower by L. E. J. Brouwer and by Brouwer’s student Arend Heyting. Brouwer refers to Intuition as a time-bound process evolving from the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory.49
Unlike Descartes’s clear and distinct ideas, once constructed, the mathematical ideas arising in this moment remain alive in mind and memory. It follows from this that the scientific observation of nature’s regularity is an effect of linking things and events in time sequences constructed in this upsurge of the past in the present. This means that nature’s regularity is a mind-made structure derived from the time relations of subjects and not an objective characteristic embedded in the natural world.50 And as for this notion of causality, so also for truth: “The whole of the Subject’s constructive thought-activity, past and present, constitutes mathematical reality and mathematical truth.”51 In other words, the “truth” and the “noncontradictory” nature of mathematical formulations are found in constructions that are the result of temporal, intuited thought activities.52
Among Intuitionist logical axioms, the Principle of Excluded Middle is called the most flawed and obvious misstatement of fact. The Principle of Excluded Middle, we recall, states that for a given proposition “P,” either “P” or “not-P” must be true.53 However, for the Intuitionist, mathematical statements, whether affirmative or negative, express the completion, in a temporality that “flows with the flux,” of the mathematician’s own inner time. Insofar as a statement may be neither true nor false currently or may turn out to be true or false at some unspecified time, Intuitionism is the expression of an open future and unpredictable free choices, for there is no a priori fixed determination of the elements of a proof.54 What this implies is (p.229) that the thought process is creative, that time is nonatomic and does not consist of infinitesimally divisible points next to one another on a line, each of which negates the previous one, but is more like a “fluid paste from which points cannot be picked out with atomist accuracy.”55 The mathematician is always able to choose how to construct any given sequence; thus every element in a construction has an indeterminate future.
This structure, with possibly true or false statements, yet an open and indeterminate future, does not admit binaries. For this reason it appears commensurate with Beauvoir’s conception of ambiguity. The chief Intuitionist objection to the excluded middle is that mathematics, like all thought, is a mental activity, so the possibilities of thought are open and unpredictable and cannot be reduced to a finite set of rules laid out in advance.56 As the creative act of an individual will, mental activities cannot be reduced to an a priori language. Language is a product of consensus, making social organization possible, and no one can halt the ability of any language user to reinterpret language. Since the social and cultural contexts are formative but never determining, the unexpected is always possible.57 This appears to be Beauvoir’s position as well, the future is open and unpredictable, not because it negates the past, as the Law of Excluded Middle demands, but on the contrary, because its logic is one that brings the past along into the present and the future.
8.5. Choosing and Willing
For Simone de Beauvoir, as for the Intuitionist, choosing and willing take time. As Beauvoir argues, it is not the moment of choice, the atomistic point wherein the past is negated, but the course of time that creates freedom. Only in the course of time can a genuine goal be pursued and only in the course of time can freedom confirm itself, (p.230) manifesting itself in its outcomes. In The Ethics of Ambiguity, she states that we escape the absurdity of the clinamen, the limit, negation, by escaping the absurdity of a pure moment. The clinamen is a physical concept, which Sartre, following Lucretius, conceives of in relation to free will or change. Rather than following a determinate path prescribed by a linked causal sequence of events, the clinamen consists of atoms that swerve contingently, originating a new movement that appears to “snap the bonds of fate, the everlasting sequence of cause and effect.”58
For the theory of the clinamen, nature runs its course toward an equilibrium, but deviations appear stochastically. This is the original determination of the direction of movement of the atom expressing the idea of looking for numerical and geometric patterns in a process of successive approximation.59 Unable to be predicted precisely, they swerve forth as minute, infinitesimal deviations from equilibrium, negating the past and producing something new in place of the past.60 As such, this description appears to conform to the very idea of negation and transcendence, to the Law of Excluded Middle, and to the idea that we are beings who make ourselves a lack of being, a nothingness, in order that there might be being.
Utilizing this idea existentially, to exist, to be a being, is to escape the causal determination of the past, to deviate moment by moment, from nothingness to nothingness, in which each new moment breaks completely with that past and so can be defined as nothingness. In this sense, to exist is also to deviate from natural forces. This means that existence or being could escape the external forces that determine it, whether society or nature. If this were to be the nature of time and change, “We do not exist . . . except through and by this deviation from equilibrium. Everything is deviation from equilibrium, except Nothing. That is to say, Identity.”61
Yet, as Beauvoir makes clear throughout her explicitly temporal philosophy, novels, and memoirs (the point of a memoir is to (p.231) bring the past into the present), the Sartrean model of the clinamen is absurd because no existence can really found itself moment by moment, that is, point by point.62 Moral freedom requires a past and a future, so, to keep the accomplished act from being just an opaque and stupid fact, we may justify it, not as a unique, atomistic decision, but as belonging to the temporal unity of our current project. When a project is complete, the value of this provisional end will be confirmed indefinitely only insofar as it too is acted upon, thereby becoming the starting point for another project. This is how creative freedom develops without congealing into facticity.63 Leaning on anterior creations, embracing the past, one creates by placing one’s confidence in future freedom.
8.6. Formalism and Philosophy
Intuitionistic logic and thinking was strongly opposed by a dominant twentieth-century trend called Formalism, which upholds the Law of Excluded Middle, using it to undo philosophy. The mathematician David Hilbert is widely considered to be among the most important sources of the Formalist tendency in contemporary thought.64 The claim has been made that, because of the vast expansion of mathematical knowledge, mathematicians themselves took on the role once inhabited by philosophers like Immanuel Kant. This occurred as concept formations rose to higher levels of generality and as conceptual abstractions and systematic fundamental ideas undid the notion of meaning.65
Moreover, to the detriment of the Kantian approach, any notion of an a priori spatial intuition became much less relevant to geometry.66 Unexpectedly, Hilbert’s Formalism has been taken to be one of the key sources of postmodern philosophers who sought to return Continental philosophy to a position of respect and autonomy. For (p.232) postmodern philosophy, as for Hilbert, only the formal, structural relations among signifiers are of interest, and what they signify can be anything as the signifier–signified relationship is arbitrary.67 Signs are immediately graspable finitary objects, that is, they are a finite number of symbols and propositions that are foundational, along with rules of inference, regardless of semantics.68
The Formalist system has certain guarantees. It is a consistent, compatible, noncontradictory extension of reasoning; it is equally accessible to all members of the community; and, significantly, questions about meaning are irrelevant.69 In language, this means that words may be diacritical, that is, mutually determining, or they may form signifying chains in which each term refers only to other terms within the chain. And so, somewhat paradoxically, just as the thought or intuition of the finitary objects of mathematics became the source of social consensus, the minimum that a mathematician cannot deny, social consensus also became the ground of linguistic usage.
Even so, the logician Kurt Gödel proceeded to raise serious objections. Hilbert replaced the vague notion of “truth” with formal demonstrability. Gödel agrees that demonstrability is definable in Formalist mathematics and that a proof is a finite sequence of symbols of a certain kind and nothing more.70 Yet if we insist that all provable statements are true, the converse, that all true statements are provable, is certainly not the case; thus there are supposedly true mathematical statements that are not provable. The conclusion is that “either mathematics is false, or there are true mathematical statements that are not provable (in a chosen formalization). This is usually referred to as ‘incompleteness.’ ”71 Postmodern philosophy, for the most part, has abandoned any concept of truth in favor of the proposition that meaning is in the method, and the method continues throughout history, if not beyond. Postmodern philosophers have thereby endorsed incompleteness, thus in a manner (p.233) asserting ignorance rather than knowledge, and formal properties rather than temporal flux or flow, as well as signifying chains rather than truth.
8.7. Irigaray: Materialism and Language
All of this leads us back to where we began with the question of the nature of Irigaray’s ontological stand. Is Irigaray in tune with Beauvoir’s phenomenological ambiguity characterized by Intuitionism? Or is Irigarary’s a postmodern philosophy and, if so, of what type? Or is it something else again? Answering this question takes us to several key texts including “Human Nature is Two,” in I Love to You, and “Is the Subject of Science Sexed?” and it returns us to “The Mechanics of Fluids,” as well as to “The Question of the Other,” where Irigaray comments briefly on Beauvoir.
In the latter text, Irigaray approves of Beauvoir’s identification of women with the other of the masculine subject, but goes on to say that, for Beauvoir, women wish to be man’s equal or similar to him, a return to the historically masculine subject.72 Given what has been previously set out, this is unfortunate. It does not take into account Beauvoir’s logic and ethics of ambiguity, the stepping away from binaries and from the Law of Excluded Middle. Nevertheless, Irigaray utilizes this criticism to state her opposition to the cultural and intellectual negation of an/other woman, as well as to state that, for her, the question of the other has been poorly formulated in the Western tradition. The other has been taken to be an other of the same rather than an/other subject, irreducible but sharing equivalent dignity.73 Let us try then to discern both Irigaray’s criticism of the Western tradition and her solution. We seek to do this in a manner commensurate with our account of Beauvoir, that is, by examining the logic underlying Irigaray’s thought.
(p.234) In her account of Irigaray’s critique of formal logic, Marjorie Hass has pointed out, that while logicians focus on only the formal properties of logical concepts, Irigaray’s perspective includes the Imaginary meaning of logical connectives, meaning the science of psychoanalysis. Let us begin by noticing that when Irigaray writes about natural science, she writes about the language of natural science, a language in which the words I, you, or we never appear.74 What does appear, what counts in natural science, is what is true or false, verifiable or falsifiable, formalizable or ambiguous, empirical or metaphysical, axiomatizable or not. The language of science proceeds as if no one, no subject, is speaking.
Perhaps for this reason, the characteristics of this language are quite specific. An imperceptible model is projected onto the world; the model is rigorously foreign, so as to enhance its objectivity; the model’s visibility is available only to a distant and largely surreptitious subject; the model’s imperceptibility is an effect of a mediating instrument; the model is ideal, meaning independent of psychical and physical aspects of the producers, thus ideally formalized; the model is universal and constitutes a unique, total world; its universality is the result of protocols agreed upon by at least two identical subjects; all of which leads to the conclusion that the discovery is useful, exploitable, and constitutes progress.75
The language in which all of these transactions take place must itself be of a certain type. It is well written, thus reasonable; it is expressed in symbols or letters that refer to no objects in the real world. It consists of an agreed-on list of symbols utilized to express common patterns of reasoning: + for or; = for equivalence and substitution; ∈ in set theory, indicating that one object is an element of or belongs to a type of object.76 Then, there are the quantifiers, the universal (All x) and the existential (some x), as well as logical connectors: negation (P or ¬P), conjunction (p · q), disjunction (p ∨ q), implication (p ⊃ q), and equivalence (p = q).77 Syntax, the order of (p.235) words, is governed by the Law of Excluded Middle, which constitutes binaries (P or ¬P is true but only one is true) and thereby also guarantees identity and the Law of Noncontradiction (not both P and ¬P).78 Together these logical laws minimize, if not eliminate, the possibility of ambiguity, ambivalence, and polyvalence.79
In spite of the claim that such symbols, connectors, and syntax are objective, Irigaray insists that they are “nonneutral,” but that this can be uncovered only by examining what is at stake for science in its research at each specific historical moment. For example, the science governing Freudian psychoanalysis is thermodynamics, specifically the first two principles of thermodynamics. Classical thermodynamics studies structures of decreasing complexity—machines that lose the capacity for work—whereas nonequilibrium thermodynamics studies entities, including living beings, which increase their complexity and gain a capacity for work.”80 The distinctions between closed and open systems are a manifestation of the conflicts between these two types of thermodynamics. Classical dynamics postulates and studies systems (like that of Freud’s scientific psychology) that are closed, if not isolated, and shut off from material flows so the systems break down. Nonequilibrium dynamics postulates and studies systems that are open to material flows and energy.
If the psyche is a closed system, its limited material and energy flows must be redirected. Sexual activity is taken to be dispersal, whereas its inhibition allows at least for constancy. Not surprisingly, this perspective is forced to the conclusion that the psychophysiological forces operating for every human individual manifest a duality, if not a contradiction: one orientation toward serving one’s own purposes and one in which one is an involuntary link in a chain.81 The open systems allowing new flows of matter and energy were given the name dissipative structures to indicate that dissipation can play a constructive role in the formation of new states insofar as they grow more complex by exporting entropy into their surroundings.82
(p.236) Similarly, according to Irigaray, the modern mathematical sciences are concerned with closed and open spaces (sets) and with infinities both large and small, but very little with what may be partially open, with wholes that are not clearly delineated, with borders, passages, and fluctuations between thresholds of wholes.83 Let us try to approach Irigaray from this point of view. If Irigaray seeks but fails to find elements essential to sexual difference in the mathematical sciences, specifically in Formalism, what would she propose in its place?
In “The Mechanics of Fluids,” Irigaray argues that women diffuse themselves according to modalities scarcely compatible with the framework of the ruling symbolics.84 This has to do not only with the lag on the part of science in elaborating a theory of fluids, but also with an internal contradiction or logical disjunction [aporia] in mathematical formalization.85 So indeed there is an issue in mathematical formalization that has to be taken up. There is no question that, for natural science, as much as for psychoanalysis, physical reality resists adequate symbolization, and its features have been idealized, smoothed over, in order to be characterized at all. Logical languages idealize; empirical reality resists adequate symbolization. Yet Irigaray remains interested in formalization and in the syntactic nature of Formalist and even Logicist theories.
8.8. The Semantics of Functional Symbols
Recall what was said in Section 8.6, that for Formalism only the formal, structural relations among signifiers are of interest and what they signify can be anything as the signifier–signified relationship is arbitrary.86 Signs are immediately graspable, finitary objects; they are a finite number of symbols and propositions that are foundational, along with rules of inference regardless of semantics.87 Irigaray does (p.237) not object to this model, and in fact, she appears to cite this approvingly, especially Göttlob Frege’s semantics of purely functional symbols, which he arrived at by borrowing from mathematics the idea that a concept is a function of a single variable whose values are truth values, that is, True or False.88
Frege proceeded by replacing the notion of a term, which can appear in either the subject or in the predicate position of a proposition, by the contrasting notions of subject and object. So, for example, “ ‘Socrates is a man,” instead of being treated as a relation between terms (SaM) is treated as the application of the concept “is a man” (Mx) to the object Socrates (s) yielding a value (a truth value) denoted by the sentence ‘Ms.’ ”89 Relations are treated in an analogous manner. The relation “is less than” (Lx,y), meaning x is less than y, utilizes pairs of objects and always has one of two truth values, True (L 3, 5 = True) and False (L 5, 3 = False).90 Concepts, in this structure, are always incomplete; they can be completed by an object, previously marked by the place-holding free variable “x,” as objects are complete self-subsistent entities.91
However, as Irigaray points out, the variable is only a variable within the limits of identity, meaning, the universal quantifier All.92 Borrowing again from mathematics, Frege postulated that since concepts can only be either True or False, quantifiers (All or some) distribute Trues and Falses over a given range of arguments. If “Mx” is the concept “is a man,” the extension of “M” is precisely that class of objects for which “is a man” is True. The truth value depends then on the existence of an object for which the concept “is a man” is True.93 Concepts with the same extension (membership) are thus identical and membership (∈ in set theory) is defined as concepts that have nonempty classes in their extension. In other words, what are called classes or sets are the extensions of concepts, mere membership. Ultimately, Frege was able to define concepts based on relations of extension, which are purely structural or syntactic properties.
(p.238) For Frege, no appeals to Intuitionism’s indeterminate future are necessary. He states that arithmetical objects, meaning numbers, are logical objects so that all true statements about these objects can be resolved by appealing to definitions derived from logical laws.94 Frege accepted that the concept of number should be elucidated by reference to sets, but he hoped to reduce the mathematical concept of sets to that of classes, which are logical objects in a pure logical theory.95 In other words, he hoped to reduce mathematics to logic. This seems to me to be what interests Irigaray. Unlike Beauvoir, for whom we have argued that Intuitionistic tendencies orient her thinking, Irigaray appears to remain staunchly Formalist or even Logicist. She states that she is interested in language as a syntactic structure in every realm of human endeavor, from mathematics, logic, and science to psychoanalysis and politics. In general, this can be characterized as a postmodern position.
Frege’s logic, we noted, is extensional, meaning he treats concepts as identical when they have the same membership in a class. Moreover, “every concept is defined over the whole universe of objects,” and any concept (Fx) can be True only if the negation (¬ Fx) of it is False. Only one can be true; thus the Law of Excluded Middle holds. As Irigaray states, “the ‘all’—of x, but also of the system—has already prescribed the ‘not-all’ of each particular relation.”96 The extension of the concept “¬ Fx” is relative to the whole universe of possible objects of the concept “Fx.” In this manner, the True concept, “Fx,” carries with it the identity and individuation of all its members, its full extension, but its negation would consist of quite heterogeneous classes or sets, and crucially for Irigaray, it cannot support identity.
For each concept, there is an object in the universe that is its extension, and this concept is also an object over and above its members, and as an object, it too must have a concept. Frege’s universe, (p.239) like a set of nesting Russian dolls, contains classes, classes of classes, classes of classes of classes, indefinitely, a strongly hierarchical and highly complex picture. As well, the universe is the universe of all possible objects, which are, however, conceived of as actual, leading to the problem, the contradiction actually, of the universe as a completed totality with deterministic membership and the universe as an object that would have to belong to itself.97
Irigaray is sensitive to the implications of this paradox. The classes and classes of classes, and so on, are, she thinks, planned for calculating and determining the truth value of each All, each universal statement of identity, as well as the All of the entire system. Given the nested nature of classes, the system cannot proceed to infinity without canceling the truth value of every class along the way, as no ultimate class ever appears, throwing them all into question. She postulates that the contradiction just cited—between the universe as a completed totality with deterministic membership and the universe as an object that would have to belong to itself (a logical contradiction)—poses the question of an unformulated “greater-than-all,” a sort of theo-logical presupposition that would support the system, a God, and its relation to a feminine “not-all.”98
The feminine “not-all” appears, claims Irigaray, in the intervals between concepts that define classes, linking them all together in “a project of exhaustive formalization, [but] already subjected to the constitution of the discourse of the ‘subject’ in set(s).”99 However, insofar as Irigaray is operating in the field of formal language systems—mathematical, logical, physical, psychoanalytic, social, and so forth—and insofar as her quest is for a language of difference, we can ask if there is a language that might be suitable to the idea that human nature is fundamentally two—not one and the negation of one—but a two that appears positively and not only as the connective, the not-all, between the sets constituting the one.
What these formal systems, all of which are syntactically equivalent, do not include, of course, are properties of real fluids, as only idealizable (thus solid) characteristics are mathematizable.100 The mathematical analysis of fluids loses the relationship to “the reality of bodies.”101 It privileges metaphor—a thing being called by the name of something that represents or symbolizes something else—over metonymy—a thing called by the name of something intimately associated with it.102 Similarity and likeness are privileged over association and the associative chain. This raises the question of ordering. In mathematics, ordering relations are fundamental. They are transitive, asymmetric, and irreflexive; thus they can be expressed in English as anything that is “–er than,” with one exception, that is, when something is “other than something else.”103 Ordering can be either discrete (point by point, each point distinct from every other) or continuous, but “it is natural to see discrete ordering as paradigmatically superlative and dense [continuous] ordering as paradigmatically comparative. The strict superlative is uniquely referring. Like that set of Russian dolls all in a row, it picks out one maximum, one tallest man, one least natural number, one first in a series, one last. The uniqueness of discrete ordering implies that for each moment, there is one and only one possible moment that succeeds it. The comparative, by contrast . . . . gives an order, but nothing more. Given any two things, it is able to place them in order, but does not of itself say how many there are in between, or whether they are next or near each other.”104 It consists in intervals that are ambiguous.
Of course, some sort of ordering is necessary to characterize time and space. Moment touching moment, contiguity, refers to discreteness. However, when one moment is contiguous to an interval—and every moment is contiguous to those intervals for which it serves as a limit, a boundary, just as every interval is contiguous to those (p.241) moments that are its limits or boundaries—this is called the continuum. This is one way of characterizing togetherness in mathematics.105 Continuous orderings, rather than focusing on unique individuals, allow for the consideration of other types of less atomistic entities and allow for the possibility of two.106
The universal, we saw, has been thought as one, but Irigaray maintains, this one does not exist. Rather, something else, something she refers to as “limit,” does exist and is inscribed in nature.107 Our logical and grammatical tools have kept us from thinking this, but we can do it, we can think the same thing that Beauvoir wanted to think: We can think two free subjects, we can think the sensible, and we can think that the natural is at least two.108 For Irigaray, the logical and mathematical role of negation leads to the acceptance of limit, but the irreducibility of each gender is the limit of each gender that is also inscribed in nature.109
We have traced the concept of limit to the temporal structure of the continuum, which is not strictly formal. It seems to correlate with the idea of metonymy, with sets that are neither similar nor superlative, but are associated in some manner. If it is correct that Irigaray is taking it up and utilizing it as a model, then it does clarify some aspects of her thought. Given that Irigaray implements her analysis in the context of formal languages, the question remains: Does she posit a model of a formal language that corresponds to her description of the woman-thing who speaks, but does not speak like or the same as any x (that is, metaphorically) but speaks fluidly (that is, using metonymy)?110
Irigaray’s proposals do not appear to reach the level of formal syntactic systems. There exist ongoing approaches to mathematics that are interested in utilizing a formal system that is nonatomistic, that would open the way to a formal system of difference. For the mathematics of topology, the chief concern is togetherness, so that topology can be used to abstract the inherent connectivity of objects (p.242) while ignoring their detailed form. Set-theoretic topology or general topology is the study of the general abstract nature of continuity or “closeness” on spaces.111 Set-theory topology is also the ground level of inquiry into the geometrical properties of spaces and the continuous functions between them, and, in that sense, it is the foundation on which the remainder of topology is grounded.112
Alfred North Whitehead attempted to develop a theory to ground topology without atomistic points, something fundamental to Irigaray’s project. Several factors affected his decision to seek a new approach. The theory of relativity abandoned the conception of space as a structured aggregate of absolute positions, serving only accidentally and indifferently as a receptacle or container of matter. Thus geometry became the science not of absolute “container” space, but of the complex relations obtaining directly between physical things. Moreover, Whitehead took the constructions of science to be expositions of the characters of things perceived, so that points and such other entities as have the same formal properties (for example, elements of time devoid of temporal extension, and “event-particles” or “instantaneous point-flashes”) are not among “the immediate data of perception, and the points of Euclid’s geometry and the mass-points and point-events of mechanics are not ‘genuine natural entities.’ ”113
Whitehead characterizes topological extensive connection by reference to the “order of Nature,” where order refers to mathematical ordered relations in hierarchical sets.114 Whitehead called these sets “abstractive classes,” abstracted from actual entities that are both spatial and temporal.115 In other words, “all actual entities are ex-tended in space and in time: . . . the points in space and in time which are used in the mathematical description of scientific data are obtained by a process of abstraction.”116 Abstractive classes converge to a limit, which might be the common boundaries two extensive magnitudes share. They might also be considered neighborhoods or open sets. However, they are still considered to constitute one whole, but in this (p.243) they might be analogous to open-system thermodynamics, whose dissipative structures indicate that the universe is a whole open to new matter and energy.117 Whitehead’s program is thought by some to have failed because it does not provide adequate structure, but other theorists remain convinced that it is viable.
In this account of Beauvoir’s logic of ambiguity as an Intuitionistic logic, we found a structure that accounts for True and False propositions in a temporality that brings the past into the present and makes possible an indeterminate future in which choice is possible. Intuitionism also utilizes a version of the temporal structure provided by the concept of the continuum; it is nonatomistic. What initially connects Beauvoir and Irigaray is that both seek a model that makes possible ambiguity, ambivalence, and polyvalence. Irigaray’s conception of the limit is a model for nonatomistic space and time, even within the framework of Formalist logic and the Law of Noncontradiction. It is simply another approach to the same problem—the quest for difference, for at least two genders and not just one and the negation of that one.
Although Beauvoir’s Intuitionistic phenomenological ambiguity and Irigaray’s Formalist language of science do not cancel one another out, their approaches cannot be harmonized in one grand scheme. Phenomenology is a situated embodied theory of cognition and emotion. Linguistics, even gendered linguistics, is a formalist, syntactical approach to science and mathematics, psychoanalysis, and politics. As Irigaray herself states, in a comment that echoes Wittgenstein of the Tractatus, outside of this structure, circumscribed by the signification articulated in masculine discourse “nothing is: a woman,” there is only a “Zone of silence.”118
(1.) Luce Irigaray, Je, Tu, Nous, Toward a Culture of Difference, trans. Alison Martin (New York: Routledge, 1993), 45; originally published as Je, tu, nous (Paris: Grasset et Fasquelle, 1990).
(2.) Tove Pettersen, “Existential Humanism and Moral Freedom in Simone de Beauvoir’s Ethics,” in Simone de Beauvoir—A Humanist Thinker, ed. Tove Petterson and Annlaug Bjørsnøs (Leiden, The Netherlands: Brill/Rodopi, 2015), 76.
(3.) Val Plumwood, “Feminism and the Logic of Alterity,” in Representing Reason, Feminist Theory and Formal Logic, ed. Rachel Joffe Falmagne and Majorie Hass (New York: Rowman and Littlefield, 2002), 45–70, 50.
(4.) Vladimir Tasić, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001), 40–41.
(5.) Jean-Paul Sartre, Being and Nothingness: An Essay in Phenomenological Ontology, trans. Hazel Barnes (New York: Washington Square Press, 1971), 33–36; originally published as L’Etre et Néant, Essai d’ontologie phénoménologique (Paris: Gallimard, 1943).
(7.) Marjorie Hass, “Negation and Difference,” Philosophy Today, Philosophy in Body, Culture, and Time, 44 (2000): 112. Hass cites Ludwig Wittgenstein, Tractatus, trans. Daniel Kolak (Mountain View, CA: Mayfield, 1998), 5.02.
(8.) Marjorie Hass, “Fluid Thinking, Irigaray’s Critique of Formal Logic,” in Representing Reason, Feminist Theory and Formal Logic, ed. Rachel Joffe Falmagne and Marjorie Hass (Lanham, MD: Rowman & Littlefield, 2002), 75–76.
(11.) Simone de Beauvoir, The Second Sex, trans. H. M. Parshley (New York: Vintage Books, 1989), 4–5.
(31.) Ibid., 495. In the United States today, celebrities are praised for their pregnant bodies—but the reality for the average woman has probably changed little. Some professions still attempt to restrict the role of pregnant women or keep them out of the public eye.
(45.) Ibid., 618.
(47.) Walter P. Van Stigt, “Brower’s Intuitionist Programme,” in From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in the 1920s, ed. Paolo Mancosu (Oxford: Oxford University Press, 1998), 1–22; 5.
(48.) Henri Bergson, Time and Free Will, An Essay on the Immediate Data of Consciousness, trans. F. L. Pogson (New York: Macmillan, 1959), 98–112. Tasić’s claim that Bergson lacks a conceptual framework is simply not the case. See my extensive account of Bergson’s ontology, “Creative Evolution: An Ontology of Change,” in Gilles Deleuze and the Ruin of Representation, (Berkeley: University of California Press, 1999), 118–146.
(49.) L. E. J. Brouwer, “Historical Background, Principles and Methods of Intuitionism,” in Collected Works, trans. Arnold Dresden (Amsterdam: North-Holland; New York: Elsevier, 1975–1976), 510.
(58.) Cited in Susan Mapstone, “Non-linear Dynamics: The Swerve of the Atom in Lucretius’ de rerum natura,” 8. Available at http://www.londonconsortium.com/wp-content/uploads/2007/02/mapstonestoicsessay.pdf. Mapstone cites Lucretius, On the Nature of the Universe, trans. R. E. Latham and revised John Godwin (London: Penguin Classics, 1994), 44.
(59.) Gilles Deleuze, Logic of Sense, ed. Constantin Boundas, trans. Mark Lester with Charles Stivale (New York: Columbia University Press, 1990), 269. Originally published as Logique du sens (Paris: Les Éditions de Minuit, 1969).
(60.) Michele Serres, The Birth of Physics, trans. Jack Hawkes (Manchester, UK: Clinamen Press, 2000), 22.
(62.) Beauvoir, The Ethics of Ambiguity, trans. Bernard Frechtman (New York: Citadel Press, 1976), 26.
(64.) Tasić, Mathematics and the Roots, 67.
(65.) Paul Bernays, “Hilbert’s Significance for the Philosophy of Mathematics,” in From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in the 1920’s, ed. Paolo Mancosu (Oxford: Oxford University Press, 1998), 189. Bernays (1888–1977) was a Swiss mathematician who worked as assistant to Hilbert. He made significant contributions to axiomatic set theory.
(66.) Bernays, “Hilbert’s Significance,” 189.
(67.) Tasić, Mathematics and the Roots, 67.
(68.) See Richard Zach, “Hilbert’s Program,” in The Stanford Encyclopedia of Philosophy, ed. Edward N. Zalta. Accessed July 2015. Available at <http://plato.stanford.edu/archives/sum2015/entries/hilbert-program/>.
(72.) Luce Irigaray, “The Question of the Other,” trans. Noah Guynn, Yale French Studies, 87 (1995):8.
(74.) Luce Irigaray, “Is the Subject of Science Sexed?,” trans. Carol Mastrangelo Bove. Hypatia 2.3 Feminism & Science (Autumn, 1987): 66.
(80.) Lynn Margulis and Dorion Sagan, What is Sex? (New York: Simon & Schuster, 1997), 32.
(81.) This model, manifest in Freud’s scientific psychology, is the subject of my essay “Catastrophe,” in Traumatizing Theory: The Cultural Politics of Affect in and Beyond Psychoanalysis, ed. Karyn Ball (New York: Other Press, 2007), 41–66.
(82.) Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Man’s New Dialogue with Nature (New York: Bantam Books, 1984), 12. Equilibrium thermodynamics studies the transformation of energy, and the laws of thermodynamics recognize that although “energy is conserved,” when energy is defined as the capacity to do work, nevertheless, nature is fundamentally asymmetrical, that is, although the total quantity of energy remains the same, its distribution changes in a manner that is irreversible. See P.W. Atkins, The Second Law (New York: Scientific American Library, 1984), 8–13. Eric D Schneider and Dorion Sagan, Into the Cool (Chicago: University Of Chicago Press, 2001), 81; and Ilya Prigogine, Thermodynamics of Irreversible Processes (New York: Wiley, 1955).
(84.) Luce Irigaray, “The Mechanics of Fluids,” in This Sex Which Is Not One, trans. Catherine Porter with Carolyn Burke (Ithaca, NY: Cornell University Press, 1985): 106.
(86.) Tasić, Mathematics and the Roots, 67.
(87.) See <http://plato.stanford.edu/archives/sum2015/entries/hilbert-program/>. Accessed July 2015.
(88.) Mary Tiles, The Philosophy of Set Theory, An Historical Introduction to Cantor’s Paradise, (London: Blackwell, 2004), 140.
(97.) Tiles, The Philosophy of Set Theory, 152–154.
(103.) J. R. Lucas, The Conceptual Roots of Mathematics (London: Routledge, 2000), 236.
(107.) Irigaray, “Human Nature is Two,” 35.
(108.) Ibid., 36–37.
(112.) Eric W. Weisstein, “Point-Set Topology,” MathWorld. Accessed July 2015. Available at http://mathworld.wolfram.com/Point-SetTopology.html.
(113.) Adolf Grünbaum, “Whitehead’s Method of Extensive Abstraction,” British Journal for the Philosophy of Science 4.15 (1953): 33.
(117.) Lucas, The Conceptual Roots, 275.