Mathematics, Infinity, and the Virtue of Humility

by Joshua Kinder

The following essay was presented at the “Virtues, Vices, and Teaching” conference hosted by the Kuyers Institute at Calvin College and is shared here with permission. 

A science which does not bring us nearer to God is worthless. But if it brings us to [God] in the wrong way, that is to say if it brings us to an imaginary God, it is worse…[1]

 “Did I get the right answer?” Students in mathematics courses ask this question, it seems, more than any other. Too often, those same students have been trained to ask about right answers by teachers who likewise obsess: “Did you get the right answer?” Classroom dialogue and assessment practices in math courses tend to be bound to the idea of the one right answer. What sort of picture does this paint of the practice of mathematics? Of mathematicians? Perhaps more importantly, what sort of world does this concern create?

This paper will introduce three thinkers along with their ideas: William Byers, a mathematician, Simone Weil, a philosopher-mystic, and Emanuel Levinas, a philosopher. After some time with each of the three, we will put their work in conversation with each other. The combination of these three will shed light on some connections between mathematics education, humility, and an ethic of nonviolence.

The light of ambiguity in mathematics

In his book, How Mathematicians Think, William Byers presents a vision of mathematics rarely seen by most non-mathematicians. His articulation of the discipline has profound implications for math educators, especially in the realm of ethical development.

Byers’s central claim is that mathematics, as a process, is a creative activity. Many of us encounter, or remember encountering, mathematics as content, as algorithmic, procedural, and cold. What is the relationship between creativity and algorithm? Can algorithms produce new ideas? Byers writes, “The creativity of mathematics does not come out of algorithmic thought; algorithms are born out of acts of creativity, and at the heart of a creative insight there is often a conflict–something problematic that doesn’t follow from one’s previous understanding.”[2] Particularly interesting in light of this paper’s subject, Byers adds, “How a person responds to the problematic tells you a great deal about them. Does the problematic pose a challenge or is it a threat to be avoided?”[3]

Three things that often characterize the problematic, in Byers’s view, are ambiguity, contradiction, and paradox. He gives considerable space to clarifying how he uses ambiguity. An ambiguous situation is one which can be understood from two self-consistent but mutually incompatible frames of reference.[4] It is the task of sorting out the ambiguous which leads to new, creative leaps in mathematics. It is this same task which leads most often to student difficulty and confusion.

To elucidate this idea of ambiguity, let’s consider an elementary example, the statement 2+3=5. On the face of it, this is a very unproblematic statement, very much unambiguous. If we dig deeper, what do we see? Often, this simple addition is taught with something like the image of a balance, with 2+3 on one side and 5 on the other. This image implies that the two sides are the same thing, but is that all the equals sign says? The equals sign in an equation is not simply a marker of sameness or identity, but is rather a sort of bridge between two worlds. Equations are mathematicians way of writing metaphors; equations are the tie that binds the two reference frames of the ambiguous. In the case of 2+3=5, the equals sign reveals a deep connection between process and object: we read that adding 2 and 3 is the same idea as the number 5. That is, the process of adding and the result of adding are two ways of looking at the same thing. “What used to be a conflict becomes a flexible viewpoint where one is free to move between the contexts of number as object and number as process.”[5]

Byers goes on to give numerous other examples of how ambiguity, contradiction, and paradox drive mathematicians ever onward towards new creative insights. What sort of person is a mathematician? What sort of character is able to enter the ambiguous, wait there for a flash of insight, and return with the gift of new knowledge? We turn now to the second of our three friends, Simone Weil, and her idea of attention.

 Simone Weil’s notion of attention

Simone Weil was a French philosopher, mystic, and political activist in the first half of the 20th century. An important idea running through her body of writing is the notion of attention, which she connects to school studies, prayer, and interpersonal relations, all of which are of interest for the purposes of this paper. (It is also interesting to note that Simone’s brother, Andre Weil, was part of the famous Bourbaki group of French mathematicians.)

In an essay titled, “Reflections on the Right Use of School Studies with a View to the Love of God,” Weil details her vision of attention, citing scenarios specific to mathematics education. Perhaps as a result of her own practice of attention, Weil’s writing displays what one commentator has described as “a revelatory certitude and restraint.”[6] She begins her essay clearly, “The key to a Christian conception of studies is the realization that prayer consists of attention…The development of the faculty of attention forms the real object and almost the sole interest of studies.”[7]

Attention is a goal of studies completely divorced from mastery of content. The time spent attending to a difficult geometry exercise, regardless of success in solving it, has benefited the student in another, “more mysterious dimension.”[8] Attention is different than simply sustaining effort over time. This “muscular effort” we see when a teacher calls students to attention is little more that just that: the muscles of the face and brow contracting, the eyes focusing, and nothing else. “Studies conducted in such a way can sometimes succeed academically from the point of view of gaining marks…[but] such studies are never of any use.”[9]

For Weil, the sort of attention worth pursuing is founded in desire, not will power. Attention is a kind of waiting, looking for the “light” without attachment to anything in particular: “Not to try to interpret…but to look…till the light suddenly dawns.”[10] Her poetic description is worth quoting at length, “Attention consists of suspending our thought, leaving it detached, empty, and ready to be penetrated by the object…Our thought should be in relation to all particular and already formulated thoughts, as a man on a mountain who, as he looks forward, sees also below him, without actually looking at them, a great many forests and plains.”[11]

Perhaps most importantly, Weil’s is an ethical attention. In her 1942 essay “Human Personality” she expounds on the value of attention as it relates to love of neighbor. She begins with the claim that what is sacred in a person is exactly that which is impersonal, that something “that goes on indomitably expecting…that good and not evil will be done to him [sic].”[12] Weil detects in every impersonality, in every soul, a ceaseless cry to be delivered from evil: “Why am I being hurt?”[13] This cry of the afflicted can be heard only when the listener comes to the limit of her own world. Each of us, Weil says, “moves in a closed space of partial truth.”[14] In order to come to the boundary of that space and move through the wall of separation, it is necessary for one to be “annihilated,” to “dwell a long time in a state…of humiliation.”[15] This choreography happens in discourse: “To listen to someone is to put oneself in his place while he is speaking. To put oneself in the place of someone whose soul is corroded by affliction, or in near danger of it, is to annihilate oneself.”[16] Weil’s final word is to identify this listening to the afflicted as the same attention with which we attend to truth. In her words, “The name of this intense, pure, disinterested, gratuitous, generous attention is love.”[17] Thus, for Simone Weil, all school studies are closely connected to the practices of prayer, love of God, and love of neighbor. It is precisely in developing our capacity to attend to truth that we develop our capacity to attend to the affliction of our neighbor.

Emanuel Levinas and the face of the Other

In Totality and Infinity, Emanuel Levinas works with the idea of the face of the Other, crafting a vision of the ethical relation which takes into account the idea of infinity and what he calls metaphysical desire. This cluster of ideas, we will see, has a strong correlation with what we’ve already seen in Byers and Weil, and helps again to bridge the mathematical and the ethical.

One of Levinas’s main projects in Totality and Infinity is to demonstrate how the idea of infinity produces a rupture in the totality. By “totality”, he refers to the idea that “individuals are reduced to being bearers of forces that command them.”[18] Totalization is the process whereby the same and the other become one thing. To be totalized is to be encompassed, swallowed up, completely known. Levinas connects totality with the history and patterns of empire and the State, as well as with the tradition of Western thought, “where the spiritual and the reasonable always reside in knowledge.”[19]

In contrast to totality, in fact undermining and rupturing it, Levinas considers the idea of infinity. Levinas, borrowing from Descartes, is fascinated by infinity because it is the one idea whose object overflows the idea itself: our minds, thinking of infinity, cannot begin to contain it. The infinite cannot be totalized. The source of the idea of infinity for Levinas is the face of the Other. “The idea of infinity, the infinitely more contained in the less, is concretely produced in the form of a relation with the face. And the idea of infinity alone maintains the exteriority of the other with respect to the same, despite this relation.”In another place he says, “In the face such as I describe…is produced the same exceeding of the act by that to which it leads. In the access to the face there is certainly also an access to the idea of God.”[20]

The origin of ethics is the face to face encounter, within which my own spontaneity is called into question by the presence of the other.[21] This calling into question is enacted by dialogue: the face speaks, I do not merely see it. “The first word of the face is…’Thou shalt not kill.’ It is an order. There is a commandment in the appearance of the face, as if a master spoke to me…At the same time, the face of the Other is destitute; it is the poor for whom I can do all and to whom I owe all.”[22] It is through this dialogue that the face resists possession, resists totalization. The face speaks as an infinity, and as such cannot be contained.

The ambiguity of the face

Having acquainted ourselves with these three, Byers, Weil, and Levinas, let us turn to the task of braiding them together if we can. Together, the three of them may illuminate connections between mathematics, virtue, and education.

The idea of ambiguity is present in each of these thinkers. This commonality, given their three quite different projects, suggests a fruitful juxtaposition. What exactly do mathematics, attention, and ethics have to do with each other?

For Levinas, ambiguity presents itself primarily in the face of the Other. The face, in dialogue, is understood alternately as interlocutor and theme. The face speaks, and I thematize it in mind, I create a certain picture of it. Without fail, this image, the other as my theme, is shattered in subsequent discourse by the other as interlocutor. Language is thus the medium by which this ambiguity of the other is revealed, and this very ambiguity is the means by which the other, in her infinity, resists totalization.[23]

In mathematics, “the concept of infinity is inherently ambiguous.”[24] Insofar as mathematics is a kind of language, it also is the medium by which the ambiguity of infinity is revealed.

“Observe the equation 1/2 + 1/4 + 1/8 + 1/16 +… = 1. On the left-hand side we seem to have incompleteness, infinite striving. On the right-hand side we have finitude, completion. There is a tension between the two sides which is a source of power and paradox. There is an overwhelming mathematical desire to bridge the gap between the finite and the infinite. We want to complete the incomplete, to catch it, to cage it, to tame it.”[25]

What we see here with an infinite sum is, I suggest, exactly what Levinas is talking about. On the left side, the infinite is resisting our attempts at totalization. We can write the terms of the sum forever, and infinity still won’t allow us to finish it. We suggest a “theme” on the right side: the list sums to one; but for students or others inexperienced with infinity there are nagging doubts. Is it really equal to 1? Can we really say that?

Mathematics is, according to mathematician Hermann Weyl, “the science of the infinite.”[26] If mathematicians consistently and habitually study infinity, then the language of mathematics must have a great deal to say about ethics in light of how Levinas has framed the subject. Thus, it is not surprising that mathematicians write above of the desire to reach the infinite, the desire so eloquently symbolized with the equals sign. And it is also not surprising that mathematicians are well acquainted with the dissonance, discomfort, and confusion so often generated by the idea of infinity. The infinite confounds the mind, and it is the task of the mathematician to attend to the infinite despite the difficulty, despite the resistance it offers. Thus, the infinite, source of so much ambiguity and paradox, source of ethics for Levinas, is also what generates the desire which leads to the attention that Weil speaks of so often.

Infinity and humility

The resistance to totalization that infinity offers, despite and because of our best practice of attention, demonstrates to the mathematician the futility of grasping at the world, trying to control it or put it in a box. The study of infinity tends to produce humility in the student. Simone Weil’s scenario of a student struggling with a geometry exercise is useful here: it is humiliating to be unable to solve a problem. We have all felt what it’s like to get back an exam with a poor score, that red ink screaming at us the fact that we didn’t know everything. Weil writes about mistakes, “When we force ourselves to fix the gaze, not only of our eyes but of our souls, upon a school exercise in which we have failed…a sense of our mediocrity is borne upon us with irresistible evidence.”[27] Paying attention to our mistakes, paying attention despite our “I don’t know”s, this is how we can develop the virtue of humility.

If attending to our ignorance is a path to humility, then attending to the infinite as the face of the Other will be most humiliating. There is an infinity in each face, an entire world we can never know. Levinas describes the resistance of the infinity of the face to any kind of completion, totalization, or domination. “The infinite paralyses power by its infinite resistance to murder, which, firm and insurmountable, gleams in the face of the Other, in the total nudity of his defenceless eyes…”[28] Here is another curious resonance between Levinas and Weil: both of them use the destitution and defenselessness of the other as a starting point for ethical thinking, and both of them start with what the destitute other says. For Levinas, the first word from the naked face is, “Thou shalt not kill,” and for Weil, the afflicted is always crying, “Why am I being hurt?”

For the one who would practice attention, who would attend to the face of the Other, humiliation is inevitable. With enough time, gazing upon the helplessness of the other becomes a gazing at our own helplessness. The face of the Other, the idea of the infinite, is a mirror. Mathematics, as the study of infinity, the practice of gazing into the infinite mirror, is thus a way toward humility. With enough time contemplating the mysteries of infinite sums, I realize I am helpless in the face of the idea of infinity. And yet, as all three of our thinkers attest, an irresistible desire to go further always remains. Isn’t that a kind of prayer?

Mathematics and nonviolence

Finally, a brief word on the contributions the practice of mathematics can make to an ethic of nonviolence. William Byers suggests that ambiguity, paradox, and contradiction are the fuel that generates mathematical creativity. Regarding process, both Byers and Simone Weil describe a kind of waiting for insight that we call attention. This practice of attention stands in contrast to the posture of totalization described by Levinas. For Levinas the spirit of the totality is the engine of war, the movement towards annihilation despite the word, “Thou shalt not kill.” This spirit of the totality would reduce the practice of mathematics from a creative enterprise to a set of algorithms for determining correct solutions. Purging the problematic from mathematics does violence to the infinite, just as committing murder does violence to the infinity of the face of the other. I submit that training mathematics students to attend to the ambiguous in mathematics, rather than training them solely in the use of algorithms, will lead to their formation as people who would rather seek multiple ways of understanding a conflict than foreshorten dialogue in the interest of their one “correct” solution. Mathematics education can serve the purposes of conflict resolution through the cultivation of attention.

 Possibilities for mathematics education

 To conclude, I will return to the question at the start of the paper, “Did I get the right answer?” There is a model of mathematics pervasive in elementary and secondary education today which presents the subject solely as a set of procedural knowledge, a collection of algorithms to be memorized, rehearsed, and mastered. Mathematics, as far as many students, exam writers, and teachers are concerned, comprises a whole lot of algorithms, some simple, some difficult, for determining the correct answer. This sort of math education necessitates the sort of “muscular effort” Simone Weil wrote about, but does very little to inculcate a capacity for attention in learners.

The vision of mathematics I’ve set forth here places the discipline within a strand of ethical and spiritual writers who do not concern themselves with right answers. Rather, the study of the infinite confronts the learner with ambiguities, uncertainties, and open endings. If math education is to further students’ journey on the road towards a virtuous life, we must begin to make room for lessons and exercises which require attention, not just muscular effort. Students must encounter the ambiguous, the contradictory, and the paradoxical in their mathematics courses.

Beyond simply encountering these, the teacher must draw their attention to the problematic, making time for students to become confused and honoring that confusion. So often, students encounter difficult math ideas as they are already understood by a teacher or expert. “If it’s so easy for the teacher, why can’t I get it already?” or “I’ll never be as smart as _______, I’ll never understand this.” Rarely do students see math ideas presented along with their historical context. Certainly, no mathematical idea was created in a quick and smooth process: creating these ideas takes time, but we don’t show that to students. Students see the end result, the idea as algorithm, and nothing more.

Also, as Weil remarks, the practice of attention requires solitude: perhaps an emphasis on groupwork detracts in some ways from students’ virtuous development in math classes.

Above all, students should leave a math course not with the sense that the world is full of correct, complete answers to be deciphered by algorithms, but rather with the sense that the world is mysterious, infinite, and not under their control.

[1]    Weil, Gravity and Grace, p. 105.

[2]    Byers. p. 23.

[3]    Byers, p. 6.

[4]    Byers, p. 28.

[5]    Byers, p. 35.

[6]    Peta Bowden. “Ethical Attention: Accumulating Understandings.” European Journal of Philosophy 6:1, p 60.

[7]    Weil, Waiting for God (2009), p. 57.

[8]    Weil (2009), p. 58.

[9]    Weil (2009), p. 59.

[10]  Weil, Gravity and Grace (1952), p. 174.

[11]  Weil (2009), p. 62.

[12]  Weil. “Human Personality.” (1986), p. 51.

[13]  Weil (1986), p. 73.

[14]  Weil, (1986), p. 69.

[15]  Weil, (1986), p. 70.

[16]  Weil, (1986), p. 71.

[17]  Weil, (1986), p. 72.

[18]  Emanuel Levinas, Totality and Infinity (1969), p. 21.

[19]  Levinas, Ethics and Infinity (1985), p. 76.

[20]  Levinas, Ethics and Infinity, p. 92.

[21]  Levinas, TI, p. 43.

[22]  Levinas, Ethics and Infinity, p. 89.

[23]  Levinas, TI, p. 195.

[24]  Byers, p. 117.

[25]  Byers, p. 118.

[26]  Quoted in Byers, p. 113.

[27]  Weil, Waiting for God, p. 60.

[28]  Levinas, TI, p. 199.

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