**Her finalist year:** 1968

**Her finalist project**: Figuring out the algebraic properties of collections of objects

**What led to the project:** Penelope Maddy always liked math, but she got particularly excited about the subject in her ninth grade algebra class at what is now Dana Middle School in San Diego. "It amazed me that you could take those little bits of information in a word problem, translate them into an equation or two, and find out the answer," she says. "I guess it was my first real inkling of the power of mathematics."

Her teacher showed her a book on abstract algebra—the study of "number systems" in which the "numbers" are not what most people would think of as numbers at all. Maddy considered what happens if the "numbers" are sets (a collection of things, like all the solar system’s planets, or all the members of Congress). "I'm sure there was nothing of continuing interest in it," she says. But when she entered it in the 1968 Westinghouse Science Talent Search she was named a finalist, and then placed seventh overall. This was particularly exciting, because on her trip to Washington, D.C., she says, "I saw snow for the first time in my life!"

**The effect on her career:** Maddy knew she wanted to continue to study set theory. So she enrolled at the University of California, Berkeley, as a math major.

Even in high school, however, she had started to ponder the limits of what sets can be used to prove. In particular, she was interested in what we can know and cannot know about infinite numbers. In math, there isn't just one "infinity," Maddy notes. There are many infinite numbers of different sizes. To begin with, there's the size of the set of natural numbers (1, 2, 3, 4…). However, the set of real numbers (those corresponding to all the points on a line, including between those numbers), which is also infinite, is bigger than the set of natural numbers. All the different infinities can be lined up -- the smallest, then the next biggest, and so forth -- and many of the familiar operations, like multiplication or raising numbers to an exponent, can be defined on these infinite numbers.

These different infinite numbers also present some perplexing problems: For instance, what happens if you take the number 2 and raise it to the smallest infinite number? "The answer will have to be infinite, but which infinite number is it?" she asks. The smallest, the next smallest…? Something called "The continuum hypothesis" (CH), proposed by Georg Cantor in the 1870s, says that the answer is the second infinite number, but whether the CH is true or false cannot be proved via the normal methods, Maddy says. You can't show whether it's true "without adding some new fundamental axiom"—that is, a basic assumption that can't be founded on anything more basic. "And nobody's yet found a satisfactory way of doing that."

To Maddy, this didn't seem like just a straightforward math question. She started to wonder: How do you justify a fundamental mathematical assumption? "You can't prove it, because it's the place where all proofs begin. So what do you do?"

When she started musing on questions like this, "it was the beginning of the end for me," she says. "I was headed into philosophy." She went to Princeton University to earn a PhD in the topic, receiving her degree in 1979. (Her thesis looked at the continuum hypothesis.) She began teaching at the University of Notre Dame and then at the University of Illinois at Chicago.

**What she's doing now:** Since 1987, Maddy has been a professor of logic and philosophy of science and of mathematics at U.C. Irvine, where she continues to ponder—and teach—the CH and other questions. She combines history and set theory in her work, and is a "highly respected and influential philosopher," says Donald A. "Tony" Martin, a set theorist and philosopher of mathematics at U.C.L.A.

She's built up a reputation for clarity—rare in the usually abstruse academic fields of math or philosophy. "In particular, she can make very technical material accessible to nontechnical readers," Martin says. "Her writing style is simple, clear and a pleasure to read."

Her philosophical views have changed over the years. For example, in 1990 she wrote *Realism in Mathematics*, which Martin calls "a very fine book, defending with great resourcefulness a realist view about mathematical objects and mathematical knowledge." (Realism is the idea that math exists independently of the human mind; we do not invent it, we discover it). But in 1997, after further contemplation of these questions, she wrote a book called *Naturalism in Mathematics*, which, in part, argues against the realist perspective. "She is not dogmatically attached to her views, as her switch from realism to naturalism dramatically shows," Martin says. "But she doesn't just hop around from one view to another. There's real sense in which her philosophical ideas have been steadily developing over time. Parts of earlier positions have been dropped, but much has been kept or adapted."