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When Malevich painted a black square, he revolutionized painting. But had he painted the first abstraction, or had he painted a square? He had painted geometry and nothing more, but was he actually doing geometry? When Terry Winters calls a series Knotted Graphs, is he illustrating topology or doing topology?

I owe the question to an artist that, reasonably enough, works in abstraction. Puzzling it out takes knowing what one means by doing mathematics. If that were not enough, it takes knowing what it means to call something abstraction or illustration. A show that claims to connect art and math may not, even as it involves both artists and mathematicians. Alyson Shotz may or may not, but her waveforms depend on both. As for Winters, it is a knotty problem.

Of course, Kazimir Malevich was not the first artist to turn to geometry. It is hard to imagine Jan Vermeer in The Art of Painting without his maulstick—or the Renaissance without a compass and straightedge. It is hard, too, to imagine any number of paintings without preparatory drawings gridded for transfer. Studio training had long required the tools, ideas, shapes, and symmetries of math along with a steady hand. Leonardo filled his notebooks with them, and indeed artists have probably found the golden rectangle and the catenary more interesting than mathematicians. But were they doing mathematics?

Art and nature may go together, but art and science intersect in curious ways. Art and math may seem even more alike, give or take the artist's skill, a free and creative hand, or the pleasure and pain of expressionism. Today, software practically builds all these right in along with the math. Yet that still leaves a huge divide. One can see it in majors and graduate programs in art as opposed to mathematics. Guess which discipline, for example, has welcomed more women?

Artists who borrow images from science or medicine are obviously not doing science or medicine—not even when they understand, respect, and do not distort the original purpose of the image. They are plainly not doing research. But they are also not doing science in the sense of a student doing homework by using the tools of science, such as math, to solve problems or by going into the lab. There are often parallels in art to what scientists do, such as close observation, creative representation, and generalization about observations. But even those do not necessarily require images from science, and the parallels take one only so far. When artists take science as subject matter, they often look to something older than the rigors of science anyway, in natural history.

However, the case of math is much less clear, since there the very distinction between sign and the thing itself is hard to pin down. Is Terry Winters doing math? Set aside his knots for now. What they are and how to represent them are an added complexity—and an unnecessary one. I hardly understand topology myself, and I had more than four years of college math and physics. Take instead something more familiar, a square. One can always ask later when knots qualify the conclusions.

It helps to realize that behind the question lie two quite distinct puzzles, although they do overlap. One puzzle has nothing at all to do with art: when one commits a square to paper, is one doing math—or even drawing a square? The second puzzle is whether it matters when the shape is a work of art. I cannot resolve either puzzle, especially the first. It has surely been around as long as civilization. But it is crucial to be aware of the question—and why it is distinct.

Logically, a drawing cannot possibly be a square. On the one hand, it has features irrelevant to a square, such as a color and the width of the lines. On the other hand, it lacks the essential features of a square. The angles are never exactly 90 degrees, and the sides are never exactly equal in length. People have responded to this puzzle in many ways over time. Their responses are in fact the philosophy of mathematics.

Some have spoken of a square as an *idealization*—in effect, a more perfect drawing than just happens to be possible. They have called a square a *concept*, existing in one's head but sharable. They have described squares as *ideals*, existing but in some other realm. For Plato even the mental pictures, much less the drawings, are just shadows or representations of reality. Or they have seen mathematics as a formal system, built on definitions and axioms. In this solution to the puzzle, a square is a *sign* in a language or is the same thing as its definition.

These are all good answers, and all have serious drawbacks. The meaning of an idealization is hardly clear, especially since by definition it is not achievable. As for mental constructs, they make math sound rather arbitrary, just when one wants mathematical truths to be universal truths—or at least true by definition. Formal models run up against the limits of all systems. Long before Postmodernism, Alfred Whitehead and Bertrand Russell sought to construct math from first principles, using set theory. Their early twentieth-century project ran up to their dismay against Gödel's paradox, which showed that they could never encompass all of mathematics without also encompassing contradictions.

Plato's solution has limits to its appeal, too. The nature of his other realm frankly eludes me. So does how he expects human beings to have access to it—unless he allows representations and formal systems back in after all. Plato also worked a long time ago, before what tortured students know today as algebra. He and his contemporaries took for granted that mathematics was geometry, not something best specified by formulas with symbols. In his time, the puzzle was the relationship between a square and a real or mental picture. Today the puzzle is the relation between an algebraic representation and a geometric one.

In fact, if anything sets apart mathematicians from us common folk, it is the ability to see the relationship as a matter of course. A knot is something governed by a set of formal rules, called mappings or transformations. However, topologists have no trouble understanding what that has to do with the intuitive idea of a knot. They know quite well that they are describing something distinguished by how it gets tied up. But then, that makes them mathematicians. A mere mortal has to settle for picturing physical reality, which is why I majored in physics and not math.

Plato's solution also runs up against a contrary intuition: of course the thing on paper is a square. I just drew one! This intuition is hardly rigorous, but it gets at the ordinary motivation for studying these things. It also gets at how real people do math—and why artists may claim to be doing math after all. Students prove things in school by drawing and labeling pictures. Later, in college, they may prove things by manipulating symbols, but it is a little discomforting then to think that they are manipulating only representations and not the thing itself.

Kurt Gödel was a Platonist, but most contemporary philosophers who have skewered formal systems are closer to pragmatists or "wholists." They are less and less likely to recognize the distinction between truth by experience and truth by definition. It all just goes into the big pot and gets stirred. Perhaps the most influential recent philosopher of logic, W. V. O. Quine, started out impatient with everyone's arguments but his own. One of his first projects was to show that the three-volume Principia could be brought off in two hundred pages. In time, he showed instead one more reason that it could not be brought off at all.

The first puzzle, then, comes down to whether the square in front of one's face is a definition, a representation, or the thing itself—or whether there is even a distinction to be made here. That applies to a square as a set of words, a picture, *or* a set of algebraic formulas. You and I are not going to resolve this, but it does help move the story along. You can already see why math in art is going to be different from science in art. You can see why we are tempted to think of pictures as not just illustrations but, yes, actual mathematics. "No ideas but the thing itself," as the poem goes, and what is art if not the thing itself?

That brings me naturally to the second, distinct problem: what changes when the pictures are art? On the one hand, it seems impossible that anything can have changed. A picture is just a picture, to Plato's dismay, but it is still a picture. In fact, as Andy Warhol or Jasper Johns makes clear, a copy of a picture is a picture, too. On the other hand, it seems obvious that everything has changed.

A high-school student draws a square to help show, say, that the opposite sides are parallel. An artist, in contrast, draws a square for its own sake—or for the sake of art. Winters is not trying to show anything about knots, or if he is, he has failed to teach me. For all I know, he could have got the math wrong. He would *still* have made his point about painting or even about knots. He would still be in the tradition of Wassily Kandinsky, describing modern art as point, line, and plane.

In other words, he would still be opening up art to the beauty of drawing, space, networks, connections, transformations, nature, and ideas. Like an artist working with images from science, he is using symbols from other disciplines than art as metaphors to make a point about painting. Is he doing math? And who cares? He is still putting math to good use and not just throwing around pretty pictures. And, somehow, that fact makes his pictures art, and it helps make him a model for the present, when many artists combine imagery and other media with abstraction.

That brings the story to a final question—and this question really does matter to art. I can now ask whether it makes a difference *how* a painter or conceptual artist works with a knot or a square. If the artist is using geometry to construct a painting, like so many since the early 1960s, it feels more like *doing* math than if the artist shows a square as an image floating in a field, like a character in a graphic novel. I could argue that Winters is doing the latter. It is why I admire him but do not count him among my favorite artists. I put him uncomfortably between the people before him who seemed to hate imagery, like Frank Stella or Brice Marden, and the artists after Winters who have taught me how to enjoy made geometries like his after all, like Julie Mehretu.

Unfortunately, this is not a hard-and-fast distinction either, a lesson of lots of postmodern art. Elizabeth Murray and Jennifer Bartlett have both managed to use geometry in both ways, as composition and subject matter, in the very same work. Call them illustrators, philosophers, painters, conceptual artists, or just artists. It hardly matters. Art would be far less rich without them. To return to where I began, though, mathematics would *not*.

When Malevich painted a square, he set a high standard for both abstraction and for geometry in art. "Measure for Measure" once again connects art and mathematics—and surprisingly much of it is not abstract. It says something that the title quotes Shakespeare rather than, say, calculus. Is the show doing math or using math only as illustration? Sometimes both, and sometimes neither one. These artists have stories to tell, and not all their stories are about geometry.

Most are, however. Only Rosaire Appel includes equations—and those simply samples out of textbooks. They could illustrate a student's fear of math, the apparent subject of her cartoons. Graphs turn up in drawings by Anne Gilman, and Cherry Pickles spins out curves from allegedly the fourth dimension high on the wall. Susan Kaprov assembles jigsaw puzzles. Most, however, let a simple forms take their own course.

They include professors of mathematics or its history, who often seem to playing around while waiting for class. Julian Voss-Andrae and George Hart both sculpt polyhedra, Hart's with additional ripples along the edge. (Ah, those pesky higher dimensions.) Folds enter often, but less as the unfolding of art's very presence, as with Dorothea Rockburne, than as decorative patterning. Chris Palmer's folds open like flowers and Martha Lewis's as something akin to mappings, while Daina Taimina crochets hers. For Thomas Parker Williams, the work unfolds as an artist's book of aluminum or wood and stained glass.

The sense of self-replicating pattern comes closest here to actual math or the philosophy of math. Both Sarah Stengle and Erik and Martin Demaine generate the paradoxical outcome of simultaneously *yes* and *no*. Kristoffer Myskja lets a machine do the job, churning out holes punched in paper with their own collective geometry. Does it all feel more rigorous than a plain black square? Does it experience the collision of simple objects and visual splendor more than Tara Donovan in Mylar? I doubt it, and I may want the rigor of the 1970s back, but I shall just have to wait.

Alyson Shotz calls her show "Wavelength," and she hardly needs the 1970s with patterns this dizzying. Her waves appear not just in multiple media and multiple shapes, but in multiple manifestations within a single work. The simplest is the most intricate, a wavelike weave of black yarn held together only by pins in the wall. Facing it, more than a thousand acrylic strips break into waveforms in three dimensions over twenty-five feet. They also provide a lesson in the wave mechanics by refracting, reflecting, and transmitting light, giving yet another meaning to the title Standing Wave. One can look at its changes again and again without quite believing that the color arises entirely from the interaction of matter and light.

Can one call the strips colorless, without misstating the relationship between mathematics and nature? (Technically, the color of a dichroic material depends on the direction in which light passes through it.) Is Shotz really doing math rather than taking physics as subject matter? It is more than a textbook illustration either way. Long after Minimalism, geometry in art is still standing. Malevich, who went to his deathbed with Black Square hanging over him, can rest in peace.

*"Measure for Measure" ran at Central Booking through April 3, 2011, Alyson Shotz at Derek Eller through March 19. I owe the question about Terry Winters to Marjorie Welish. *