by Amy Ellis, Associate Professor of Mathematics Education, University of Wisconsin-Madison
When I was 16, I did a 6-week summer program at Carnegie Mellon in which I took some introductory courses in Calculus and Physics. While there I met a friend who liked math as much as I did. As August began to stretch into September, we returned to our respective hometowns (LA for him, Spokane for me) and commenced our senior year in high school.
Throughout that year, we would send letters via cassette tape, and we
would include math problems or math puzzles for one another. One day, I
got a tape with a problem he posed for me, and I remember just how he
introduced it: "I'll be reeeeaally impressed if you can solve this," he
said. "*Really* impressed."
Here was the problem. Imagine you have a cube that is made up of tiny cubes. Take apart the cube and reassemble it into two smaller cubes, with no tiny cubes left over. What are the dimensions of the original cube?
Well, that seemed like an easy, straightforward problem. Why would solving that be so impressive? I began playing around with some perfect cubes, trying to find a solution to a^3 = b^3 + c^3.
Those of you who aren't an ignorant teenager from Eastern Washington might immediately recognize that my friend set me up. I tried and tried all night to come up with a perfect cube triple that would satisfy the equation, but I could not. I made a program on my HP calculator to run a search, but it didn't yield any triples. The next day, I went into school and made a program for the Apple IIe to run a search for a triple to satisfy the equation. By the end of the day, it was still searching, and the numbers were huge. Frustrated, I took it to my math teacher, who took one look at the problem and started laughing. "Amy," he laughed, "There's a famous theorem that states there is no solution to this equation!"
Unbeknownst to me, right at that time the mathematician Andrew Wiles was working on what would become the successful proof of Fermat's Last Theorem.
I was unfamiliar with Fermat's Last Theorem at 17, but Andrew Wiles first encountered it as a 10-year old. In 1637, Fermat conjectured a^n + b^n = c^n has no non-zero integer solutions for x, y, and z when n > 2. Fermat made an intriguing note in the margin of his manuscript: "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain."
In the early 60's, at the age of 10, Andrew Wiles was captured by the simplicity of the theorem and began to try to come up with a proof of it. Fermat had proved the case for n = 4, and Euler wrote to Goldbach in 1753 that he had a proof for n = 3. Over time mathematicians were able to prove cases for other n's, but the problem remained to prove the general case. By the time Wiles encountered the theorem as a child, mathematicians had decided to put the theorem aside because they considered a proof impossible.
In the 1997 documentary by NOVA called The Proof, Wiles explained, "This problem, this particular problem, just looked so simple. It just looked as if it had to have a solution. And of course, it's very special because Fermat said he had a solution."
For over 300 years Fermat's Last Theorem intrigued mathematicians, professional and amateur alike. The theorem's notoriety grew over time as mathematicians failed to make substantial progress in solving it, with science academies offering large prizes to whoever could prove the general case. After the mid-1800s, most mainstream mathematicians gave up on proving Fermat's theorem. Furthermore, pursuing the problem had no known implications for other areas of mathematics. This meant that pursuing a proof was a big intellectual risk. A mathematician could spend an entire career attempting a proof and come up with nothing, with little to show for that effort.
Why would this endeavor be such an intellectual risk? Wiles explained, "The problem with working on Fermat is that you could spend years getting nothing. It's fine to work on any problem so long as it generates mathematics. Almost the definition of a good mathematical problem is the mathematics it generates, rather than the problem itself." Fermat's theorem was considered useless, in a certain sense, because it had no practical value in terms of generating new mathematics. It wasn't a mainstream, central question in mathematics.
As a young mathematician Wiles put aside Fermat's Last Theorem and took up the study of elliptic curves under his advisor at Cambridge, John Coates. Wiles became a successful mathematician and joined the faculty at Princeton, gaining the position of professor in 1981. In 1986, Ken Ribet, who was at Berkeley, built on the work of Gerhard Frey to establish a link between Fermat's Last Theorem, elliptic curves, and a conjecture known as the Taniyama-Shimura conjecture. Ribet's findings showed that in order to prove Fermat's Last Theorem one only had to prove the Taniyama-Shimura conjecture. Wiles knew that from that moment, that would be the problem he worked on.
Under the protection of tenure, Wiles abandoned all of his other research. He cut himself off from the rest of the world, and for the next seven years, he worked on proving the Taniyama-Shimura conjecture, and consequently Fermat's theorem, most of that time spent in complete secrecy.
Ribet noted, "I was one of the vast majority of people who believed that the Shimura-Taniyama conjecture was just completely inaccessible, and I didn't bother to prove it - even think about trying to prove it. Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you could actually go and prove this conjecture."
Wiles worked in secrecy and isolation because talking to people about Fermat generated too much interest. His colleagues had no idea what he was working on, and thought perhaps he was finished as a mathematician. His colleague Peter Sarnak thought, "Maybe he's run out of ideas. That's why he's quiet." After 6 years of working completely alone, Wiles was close to a proof, and confided in his colleague Nick Katz, to help him verify one part of the proof. Katz found no problems and in June 1993 Wiles delivered a series of lectures at Cambridge in which he concluded by announcing that he had proved Fermat's Last Theorem.
Later, the proof was found to contain a flaw, but after a year of collaborative work with Richard Taylor, Wiles fixed the problem. Wiles and Taylor published their work in May 1995, in two papers in the journal Annals of Mathematics. The final proof ran 130 pages. Wiles was subsequently awarded the Schock Prize in Mathematics, the Prix Fermat, the Wolf Prize, and was elected to the National Academy in Sciences, receiving its mathematics prizes. His proof made history.
I have been thinking a lot about Wiles' story in the wake of Wisconsin's push to effectively end tenure. On May 29 the Joint Finance Committee introduced a motion that included broad provisions for terminating tenured faculty, which was passed in July. Moreover, in October faculty at UW-Madison discovered that Madison will not be allowed to write its own tenure policy, but instead will be bound by the parameters of a system-wide policy developed by the Board of Regents. The Regents policy will mandate a post-tenure review process, one of whose possible outcomes is termination of a tenured appointment based on unsatisfactory progress.
Many people make arguments for tenure based on the need to protect faculty in speaking out against their administration, in having the protection to question the status quo, or in conducting research that may be politically risky. These are important considerations for tenure protections and should not be ignored. But Wiles' story reminds me of another, less-often discussed reason for tenure protections: The importance of protecting intellectually risky research.
Under the protection of tenure, Wiles worked in complete secrecy for over 6 years on a problem that did not yield much in terms of publications along the way. His colleagues thought he might be washed up. What would his post-tenure review have looked like? Would Wiles have been able to show sufficient productivity to avoid sanctions and possibly termination? Would he have had the confidence to even attempt his proof under a policy that would have required regular demonstration of post-tenure productivity? Under the threat of possible termination?
Yes, most of us are not Andrew Wiles. But tenure offers the protection necessary to attempt the really big problems, the scientifically risky ideas, the ones that probably won't pan out, but might, and if they do, have the opportunity to move the field forward in historic ways. We need a system in place that enables faculty to go after the intellectual brass ring. That pushes faculty to think expansively about their research, to be creative, to take big intellectual risks. Tenure affords those monumental leaps in knowledge.
As a teenager interested in solving math problems, I had no idea that I would end up as a tenured faculty member at the University of Wisconsin. I truly believe it is a privilege to do the work I do. The directions of my research have taken me to places I could not have predicted a decade ago, and I hope that will continue to be true as my career evolves. I can think of no better career than the one I have. But I have also noticed that my work is more creative and more expansive post tenure. I am more willing to take risks, to follow a path that seems like it might lead to something fruitful, even if I'm not sure.
I hope that the UW Regents and the state of Wisconsin will eventually reverse its current course to reinstate strong, robust tenure protections. Whether it's the discovery of vitamin D by UW-Madison biochemist Harry Steenbock, or the first bone marrow transplant, performed at UW hospital, or the proof of Fermat's Last Theorem, tenure protections affords significant advancements in research in science, in mathematics, in the humanities, and in the social sciences. We want those advancements to continue to happen in the great state of Wisconsin.
Here was the problem. Imagine you have a cube that is made up of tiny cubes. Take apart the cube and reassemble it into two smaller cubes, with no tiny cubes left over. What are the dimensions of the original cube?
Well, that seemed like an easy, straightforward problem. Why would solving that be so impressive? I began playing around with some perfect cubes, trying to find a solution to a^3 = b^3 + c^3.
Those of you who aren't an ignorant teenager from Eastern Washington might immediately recognize that my friend set me up. I tried and tried all night to come up with a perfect cube triple that would satisfy the equation, but I could not. I made a program on my HP calculator to run a search, but it didn't yield any triples. The next day, I went into school and made a program for the Apple IIe to run a search for a triple to satisfy the equation. By the end of the day, it was still searching, and the numbers were huge. Frustrated, I took it to my math teacher, who took one look at the problem and started laughing. "Amy," he laughed, "There's a famous theorem that states there is no solution to this equation!"
Unbeknownst to me, right at that time the mathematician Andrew Wiles was working on what would become the successful proof of Fermat's Last Theorem.
I was unfamiliar with Fermat's Last Theorem at 17, but Andrew Wiles first encountered it as a 10-year old. In 1637, Fermat conjectured a^n + b^n = c^n has no non-zero integer solutions for x, y, and z when n > 2. Fermat made an intriguing note in the margin of his manuscript: "I have discovered a truly remarkable proof of this theorem which this margin is too small to contain."
In the early 60's, at the age of 10, Andrew Wiles was captured by the simplicity of the theorem and began to try to come up with a proof of it. Fermat had proved the case for n = 4, and Euler wrote to Goldbach in 1753 that he had a proof for n = 3. Over time mathematicians were able to prove cases for other n's, but the problem remained to prove the general case. By the time Wiles encountered the theorem as a child, mathematicians had decided to put the theorem aside because they considered a proof impossible.
In the 1997 documentary by NOVA called The Proof, Wiles explained, "This problem, this particular problem, just looked so simple. It just looked as if it had to have a solution. And of course, it's very special because Fermat said he had a solution."
For over 300 years Fermat's Last Theorem intrigued mathematicians, professional and amateur alike. The theorem's notoriety grew over time as mathematicians failed to make substantial progress in solving it, with science academies offering large prizes to whoever could prove the general case. After the mid-1800s, most mainstream mathematicians gave up on proving Fermat's theorem. Furthermore, pursuing the problem had no known implications for other areas of mathematics. This meant that pursuing a proof was a big intellectual risk. A mathematician could spend an entire career attempting a proof and come up with nothing, with little to show for that effort.
Why would this endeavor be such an intellectual risk? Wiles explained, "The problem with working on Fermat is that you could spend years getting nothing. It's fine to work on any problem so long as it generates mathematics. Almost the definition of a good mathematical problem is the mathematics it generates, rather than the problem itself." Fermat's theorem was considered useless, in a certain sense, because it had no practical value in terms of generating new mathematics. It wasn't a mainstream, central question in mathematics.
As a young mathematician Wiles put aside Fermat's Last Theorem and took up the study of elliptic curves under his advisor at Cambridge, John Coates. Wiles became a successful mathematician and joined the faculty at Princeton, gaining the position of professor in 1981. In 1986, Ken Ribet, who was at Berkeley, built on the work of Gerhard Frey to establish a link between Fermat's Last Theorem, elliptic curves, and a conjecture known as the Taniyama-Shimura conjecture. Ribet's findings showed that in order to prove Fermat's Last Theorem one only had to prove the Taniyama-Shimura conjecture. Wiles knew that from that moment, that would be the problem he worked on.
Under the protection of tenure, Wiles abandoned all of his other research. He cut himself off from the rest of the world, and for the next seven years, he worked on proving the Taniyama-Shimura conjecture, and consequently Fermat's theorem, most of that time spent in complete secrecy.
Ribet noted, "I was one of the vast majority of people who believed that the Shimura-Taniyama conjecture was just completely inaccessible, and I didn't bother to prove it - even think about trying to prove it. Andrew Wiles is probably one of the few people on earth who had the audacity to dream that you could actually go and prove this conjecture."
Wiles worked in secrecy and isolation because talking to people about Fermat generated too much interest. His colleagues had no idea what he was working on, and thought perhaps he was finished as a mathematician. His colleague Peter Sarnak thought, "Maybe he's run out of ideas. That's why he's quiet." After 6 years of working completely alone, Wiles was close to a proof, and confided in his colleague Nick Katz, to help him verify one part of the proof. Katz found no problems and in June 1993 Wiles delivered a series of lectures at Cambridge in which he concluded by announcing that he had proved Fermat's Last Theorem.
Later, the proof was found to contain a flaw, but after a year of collaborative work with Richard Taylor, Wiles fixed the problem. Wiles and Taylor published their work in May 1995, in two papers in the journal Annals of Mathematics. The final proof ran 130 pages. Wiles was subsequently awarded the Schock Prize in Mathematics, the Prix Fermat, the Wolf Prize, and was elected to the National Academy in Sciences, receiving its mathematics prizes. His proof made history.
I have been thinking a lot about Wiles' story in the wake of Wisconsin's push to effectively end tenure. On May 29 the Joint Finance Committee introduced a motion that included broad provisions for terminating tenured faculty, which was passed in July. Moreover, in October faculty at UW-Madison discovered that Madison will not be allowed to write its own tenure policy, but instead will be bound by the parameters of a system-wide policy developed by the Board of Regents. The Regents policy will mandate a post-tenure review process, one of whose possible outcomes is termination of a tenured appointment based on unsatisfactory progress.
Many people make arguments for tenure based on the need to protect faculty in speaking out against their administration, in having the protection to question the status quo, or in conducting research that may be politically risky. These are important considerations for tenure protections and should not be ignored. But Wiles' story reminds me of another, less-often discussed reason for tenure protections: The importance of protecting intellectually risky research.
Under the protection of tenure, Wiles worked in complete secrecy for over 6 years on a problem that did not yield much in terms of publications along the way. His colleagues thought he might be washed up. What would his post-tenure review have looked like? Would Wiles have been able to show sufficient productivity to avoid sanctions and possibly termination? Would he have had the confidence to even attempt his proof under a policy that would have required regular demonstration of post-tenure productivity? Under the threat of possible termination?
Yes, most of us are not Andrew Wiles. But tenure offers the protection necessary to attempt the really big problems, the scientifically risky ideas, the ones that probably won't pan out, but might, and if they do, have the opportunity to move the field forward in historic ways. We need a system in place that enables faculty to go after the intellectual brass ring. That pushes faculty to think expansively about their research, to be creative, to take big intellectual risks. Tenure affords those monumental leaps in knowledge.
As a teenager interested in solving math problems, I had no idea that I would end up as a tenured faculty member at the University of Wisconsin. I truly believe it is a privilege to do the work I do. The directions of my research have taken me to places I could not have predicted a decade ago, and I hope that will continue to be true as my career evolves. I can think of no better career than the one I have. But I have also noticed that my work is more creative and more expansive post tenure. I am more willing to take risks, to follow a path that seems like it might lead to something fruitful, even if I'm not sure.
I hope that the UW Regents and the state of Wisconsin will eventually reverse its current course to reinstate strong, robust tenure protections. Whether it's the discovery of vitamin D by UW-Madison biochemist Harry Steenbock, or the first bone marrow transplant, performed at UW hospital, or the proof of Fermat's Last Theorem, tenure protections affords significant advancements in research in science, in mathematics, in the humanities, and in the social sciences. We want those advancements to continue to happen in the great state of Wisconsin.
Thank you, Amy, for your beautiful and well-argued post.
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