The painter Agnes Martin once took a walk through a garden with her friend, plucked a rose from a bush, and asked, “Is this rose beautiful?” Hearing a response in the affirmative, Martin tucked the rose behind her back and asked, “Now, is it still beautiful?” Her friend affirmed, “Yes”, and Martin concluded, “You see? Beauty is in the mind and not the rose.” Professor Barry Mazur’s book Imagining Numbers reminded me of this anecdote in its aim to reveal in the reader’s mind an intuition for conceiving of the square root of minus one. Professor Mazur’s keen manipulation of the slippery and instantaneous act of imagining results in a provocative illumination of concepts that evade visualization.
AUDREY ZHENG: I read an essay by Louise Glück in which she writes that the sublime and transcendent subjects of TS Eliot’s poetry were motivated by his “religious mind, with its hunger for meaning and disposition to awe…”. Do you think that your childhood education in a Yeshiva similarly influenced your intellectual proclivities?
BARRY MAZUR: Very much! I have recently written about one of the rabbis in my Yeshiva and how he insisted to the class that we (meaning himself and all the students in the class) are companions in study—we’ll never know it all, or all that much. We are little mice ‘nibbling on the infinite cheese of knowledge’ was his phrase. The effort we make–in contrast to the results we might find—is what is glorious.
AZ: Do you think your education in a Yeshiva gave you a greater comfort with abstraction?
BM: It must have had an effect, promoting a poetic take on things—an invitation to a more abstract viewpoint.
AZ: Your book Imagining Numbers gave me the sense that you enjoy etymology. I also read that you co-teach a seminar at Harvard dissecting and exploring a different word every course. What is your favorite word, or what is a word that particularly provokes you, whether it’s in regards to the word’s etymology, common usage, sound, or definition?
BM: ‘Favorite’? Oh my. The answer to this question is non-static and depends so much on the moment you ask it, and the thoughts I happen to be thinking when you ask it. While teaching the course focused on the word ‘Intuition’—which I did this fall—the answer would probably have been—you guessed it—‘Intuition’. At this moment, the word is ‘Beauty’ due to a different project that I happen to be thinking about now.
AZ: In your book Imagining Numbers, you wrote that the twelfth-century mathematician Bhaskara notated unknowns in his equations with the first syllables of words for different colors, and you speculate that he did this in order to “counter the blankness of features of the various unknowns in his equations”, to make them “vivid… in his imagination.” Do you share a similar habit of mentally dressing up the extremely austere in a tangible outfit?
BM: Don’t you?
AZ: Yes, but I think the interesting thing in Bhaskara’s case was that his connection between unknowns and colors was completely arbitrary.
BM: There are many examples of imaginatively useful non-arbitrary connections being drawn, and even some connections that might be thought of as arbitrary, such as Rimbaud’s sonnet “Voyelles”, are actually not arbitrary upon a deeper analysis. His sonnet begins:
A noir, E blanc, I rouge, U vert, O bleu: voyelles,
Je dirai quelque jour vos naissances latentes:
(A black, E white, I red, U green, O blue: vowels,
I shall tell one day of your mysterious origins:)
The opening move to many algebraic discussions is ‘Let X be…’ where one identifies a completely arbitrary symbol ‘X’ with some —ostensibly unknown—algebraic concept or structure.
AZ: In your book, you muse that the words ‘yellow’ and ‘tulip’ in Ashbery’s phrase “the yellow of the tulip” mutually intensify each other’s vitality in the human imagination. Does the relationship between Analysis and Geometry in the field of Analytical Geometry exhibit this sort of two-way mutually-strengthening interaction?
BM: Precisely, and not only does it also have such a mutually-strengthening relationship, but René Descartes, who is usually viewed as responsible for Analytic Geometry, said that the fusion, so to speak, of Analysis and Geometry corrects the defects of both subjects.
AZ: I wonder what a defect of Algebra might be.
BM: Well, Geometry has the virtue of being a setting for which there can be ‘visual proofs’. That is, proofs that have the property that a diagram immediately conveys the idea behind it.
AZ: Before I read Imagining Numbers I never thought of the imagination as something that could impede our ability to maneuver freely. You describe the substitution of the evocative notation of ‘Lato’ (meaning side in Latin) with the less evocative ‘r.q.’ as freeing up the restraints wedded to the geometric conception of the square root operator. Does Analytic Geometry similarly free up Geometry by removing our imaginative conceptions from it?
BM: Analysis does not take away spatial imagination from Geometry, but adds on to it its own type of imagination. For example, describing spaces as having dimensions---(it’s called ‘Hausdorff Dimension’)---that are not whole numbers. The ‘Golden Dragon’ has dimension approximately 1.61803, as it lies in the plane (which is 2-dimensional), so it is of dimension less than or equal to 2, but it wiggles around with an ‘orbit’ that is quite exorbitant so its ‘Hausdorff Dimension’ turns out to be greater than one.
Fig 1. Golden Dragon
AZ: That reminds me of Viktor Shklovsky's literary concept that is sometimes called ‘strange-ification’. He wrote that the purpose of art is to present something mundane at a slant in order to slow down the brain’s ability to perceive it. I wonder if Analytic Geometry does something similar, sort of ‘strange-ifying’ Geometry so that we can better see possible interactions.
BM: I imagine you are also hinting at Emily Dickinson’s “Tell all the truth but tell it slant” and “The truth must dazzle gradually.” Perhaps there is also a hint of reframing—and therefore a sort of ‘strange-ification’— in ‘Scheme Theory’, thanks to which Geometry, Number Theory, and Analysis can share the same vocabulary.
AZ: Is the Algebraic Geometry we learn about in high school similar to this? Like an Algebraic expression can also describe a shape?
BM: Yes, and Scheme Theory displays the relationship on a broader scale, for example relating prime numbers to Geometry.
AZ: Now I’m thinking about the distinction between form and content, which I’m more familiar with as it's discussed in literature. It seems that in Mathematics the boundary between form and content might be somewhat blurred, like the way that something is notated or expressed is actually part of the content.
BM: In Mathematics the form becomes the subject. Take, for example, a notion like congruent triangles. To state that there is congruency between two triangles is also making a statement about the abstract concept of congruency itself, which then can become the subject.
AZ: I read that you enjoy playing piano. Which composer’s pieces do you most enjoy playing?
BM: The piano lessons I had followed my inclination for Classical music; composers such as Mozart, Beethoven, Brahms, Chopin.
AZ: It’s interesting that you mentioned Classical music, because I feel like a lot of songs from that genre are very elegant and austere, and similar in style to the poems you’ve included in your book Imagining Numbers. Do you enjoy poems that have more grit to them, like the poems by Frank O’Hara?
BM: Yes. ‘Grit’ is a nice word to use as a descriptor of poetry, because it seems that many poems are actually ‘gritty’, like they disrupt smoothness. Haiku as a form almost paradoxically combines the smoothness of image with their inherently gritty syllable pattern that limits the poem’s stretch into expansiveness, for example there is Issa’s haiku:
In this world
we walk on the roof of hell
gazing at flowers.
AZ: Can you categorize, in maybe one word, the sort of poem that speaks to you?
BM: A difficult question, but ‘surprise’ is the word that comes to mind.
AZ: Whimsy and conviction are often said to be important qualities for writing a successful poem. Are these qualities useful when engaging in Mathematics?
BM: About whimsy, game theory—after all—incorporates in its essence some aspect of whimsy. Mathematicians make use of rather interesting and imaginative leaps to explain mathematics, or to be the substance of mathematics in its own right.
AZ: And regarding conviction as an important quality?
BM: How would you define it?
AZ: Sometimes I struggle to differentiate between the definitions of ‘conviction’ and ‘courage’ but I think the distinction must be important. I think there must be something at once more spiritual and intellectual about conviction.
BM: The answer is yes. I’m thinking of examples, there are famous examples but also there are the daily examples. One of the famous examples that comes to my mind is L.E.J. Brouwer. He was a great mathematician. In one phase of his work, he proved something called the Brouwer fixed-point theorem. Do you have an idea what that might be?
AZ: I don’t.
BM: So you take a napkin, put it on a table, and see where it’s placed. It’s placed on a certain square of a table, let’s say. Pick it up, crumple it up as much as you want (but you can’t tear any of it), and splunk it down so it goes in the same place, yeah? The fixed-point theorem says that there is a point on that napkin that after you’ve crumpled it and you’ve put it down, is resting on the same point of the table that it was in the beginning.
Now L.E.J. Brouwer was celebrated for having proved that theorem… but later in his life he revised his idea of what it is that constitutes a proof. He felt that the proof that he had offered for that ‘fixed point theorem’ did not fit that new criterion. To the end of his life he rejected his own proof. Now isn’t that a striking example of conviction?
AZ: It is now 2:45 for you. At 2 o’clock Barry Mazur said his favorite word was ‘beauty’. Does 2:45 Barry Mazur have a different answer?
BM: [Laughs] Oh, no, same answer. I’m hanging onto it.
