Originally published here.

A visitor to the Paris apartment of the blind geometer Bernard Morin finds much to see. On the wall in the hallway is a poster showing a computergenerated picture, created by Morin’s student François Apéry, of Boy’s surface, an immersion of the projective plane in three dimensions. The surface plays a role in Morin’s most famous work, his visualization of how to turn a sphere inside out. Although he cannot see the poster, Morin is happy to point out details in the picture that the visitor must not miss. Back in the living room, Morin grabs a chair, stands on it, and feels for a box on top of a set of shelves.

He takes hold of the box and climbs off the chair safely—much to the relief of the visitor. Inside the box are clay models that Morin made in the 1960s and 1970s to depict shapes that occur in intermediate stages of his sphere eversion. The models were used to help a sighted colleague draw pictures on the blackboard. One, which fits in the palm of Morin’s hand, is a model of Boy’s surface. This model is not merely precise; its sturdy, elegant proportions make it a work of art. It is startling to consider that such a precise, symmetrical model was made by touch alone. The purpose is to communicate to the sighted what Bernard Morin sees so clearly in his mind’s eye.

A sighted mathematician generally works by sitting around scribbling on paper: According to one legend, the maid of a famous mathematician, when asked what her employer did all day, reported that he wrote on pieces of paper, crumpled them up, and threw them into the wastebasket. So how do blind mathematicians work? They cannot rely on backof-the-envelope calculations, half-baked thoughts scribbled on restaurant napkins, or hand-waving arguments in which “this” attaches “there” and “that” intersects “here”. Still, in many ways, blind mathematicians work in much the same way as sighted mathematicians do. When asked how he juggles complicated formulas in his head without being able to resort to paper and pencil, Lawrence W. Baggett, a blind mathematician at the University of Colorado, remarked modestly, “Well, it’s hard to do for anybody.” On the other hand, there seem to be differences in how blind mathematicians perceive their subject. Morin recalled that, when a sighted colleague proofread Morin’s thesis, the colleague had to do a long calculation involving determinants to check on a sign. The colleague asked Morin how he had computed the sign. Morin said he replied: “I don’t know—by feeling the weight of the thing, by pondering it.”

Blind Mathematicians in History

The history of mathematics includes a number of blind mathematicians. One of the greatest mathematicians ever, Leonhard Euler (1707–1783), was blind for the last seventeen years of his life. His eyesight problems began because of severe eyestrain that developed while he did cartographic work as director of the geography section of the St. Petersburg Academy of Science. He had trouble with his right eye starting when he was thirty-one years old, and he was almost entirely blind by age fiftynine. Euler was one of the most prolific mathematicians of all time, having produced around 850 works. Amazingly, half of his output came after his blindness. He was aided by his prodigious memory and by the assistance he received from two of his sons and from other members of the St. Petersburg Academy. The English mathematician Nicholas Saunderson (1682–1739) went blind in his first year, due to smallpox. He nevertheless was fluent in French, Greek, and Latin, and he studied mathematics. He was denied admission to Cambridge University and never earned an academic degree, but in 1728 King George II bestowed on Saunderson the Doctor of Laws degree. An adherent of Newtonian philosophy, Saunderson became the Lucasian Professor of Mathematics at Cambridge University, a position that Newton himself had held and that is now held by the physicist Stephen Hawking. Saunderson developed a method for performing arithmetic and algebraic calculations, which he called “palpable arithmetic”. This method relied on a device that bears similarity to an abacus and also to a device called a “geoboard”, which is in use nowadays in mathematics teaching. His method of palpable arithmetic is described in his textbook Elements of Algebra (1740). It is possible that Saunderson also worked in the area of probability theory: The historian of statistics Stephen Stigler has argued that the ideas of Bayesian statistics may actually have originated with Saunderson, rather than with Thomas Bayes [St]. Several blind mathematicians have been Russian. The most famous of these is Lev Semenovich Pontryagin (1908–1988), who went blind at the age of fourteen as the result of an accident.

His mother took responsibility for his education, and, despite her lack of mathematical training or knowledge, she could read scientific works aloud to her son. Together they fashioned ways of referring to the mathematical symbols she encountered. For example, the symbol for set intersection was “tails down”, the symbol for subset was “tails right”, and so forth. From the time he entered Moscow University in 1925 at age seventeen, Pontryagin’s mathematical genius was apparent, and people were particularly struck by his ability to memorize complicated expressions without relying on notes. He became one of the outstanding members of the Moscow school of topology, which maintained ties to the West during the Soviet period. His most influential works are in topology and homotopy theory, but he also made important contributions to applied mathematics, including control theory. There is at least one blind Russian mathematician alive today, A. G. Vitushkin of the Steklov Institute in Moscow, who works in complex analysis. France has produced outstanding blind mathematicians. One of the best known is Louis Antoine (1888–1971), who lost his sight at the age of twentynine in the first World War.

According to [Ju], it was Lebesgue who suggested Antoine study two- and three-dimensional topology, partly because there were at that time not many papers in the area and partly because “dans une telle étude, les yeux de l’esprit et l’habitude de la concentration remplaceront la vision perdue” (“in such a study the eyes of the spirit and the habit of concentration will replace the lost vision”). Morin met Antoine in the mid-1960s, and Antoine explained to his younger fellow blind mathematician how he had come up with his best-known result. Antoine was trying to prove a three-dimensional analogue of the JordanSchönflies theorem, which says that, given a simple closed curve in the plane, there exists a homeomorphism of the plane that takes the curve into the standard circle. What Antoine tried to prove is that, given an embedding of the two-sphere into three-space, there is a homeomorphism of three-space that takes the embedded sphere into the standard sphere. Antoine eventually realized that this theorem is false. He came up with the first “wild embedding” of a set in three-space, now known as Antoine’s necklace, which is a Cantor set whose complement is not simply connected. Using Antoine’s ideas, J. W. Alexander came up with his famous horned sphere, which is a wild embedding of the two-sphere in three-space. The horned sphere provides a counterexample to the theorem Antoine was trying to prove. Antoine had proved that one could get the sphere embedding from the necklace, but when Morin asked him what the sphere embedding looked like, Antoine said he could not visualize it.

Mathematics Accessible to the Blind

It is easy to understand how well-meaning people who know little about mathematics might assume that the subject’s technical notation would create an insurmountable barrier for blind people. But in fact, mathematics is in some ways more accessible for the blind than other professions. One reason is that mathematics requires less reading because mathematical writing is compact compared to other kinds of writing. “In mathematics,” Salinas noted, “you read a couple of pages and get a lot of food for thought.” In addition, blind people often have an affinity for the imaginative, Platonic realm of mathematics. For example, Morin remarked that sighted students are usually taught in such a way that, when they think about two intersecting planes, they see the planes as two-dimensional pictures drawn on a sheet of paper. “For them, the geometry is these pictures,” he said. “They have no idea of the planes existing in their natural space.” Because blind students do not use drawings, it is natural for them to think about the planes in an abstract way.

The most famous blind American mathematician right now may be Zachary J. Battles, whose extraordinary story was even covered in People magazine. Blind almost from birth and adopted from a South Korean orphanage when he was three years old, Battles went on to earn a bachelor’s degree in mathematics and a bachelor’s and master’s in computer science from Pennsylvania State University. He also traveled to Ukraine twice to teach English as a second language and worked as a mentor for other disabled students. He is now studying mathematics at the University of Oxford on a Rhodes Scholarship. Like so many other blind mathematicians, Battles is an inspiration to the sighted and the blind alike.

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