Skip to main content


Chris Holden,


“You can’t prove a negative!” You probably hear it all the time. But in math, we prove negatives for breakfast. In fact, the impossible has been a driving force like no other in this most exact of disciplines.

We all know a little bit about the impossible in math: why is there is no highest number to which you can count? Because you can always add one more. It’s a little bit harder to show that there is no last prime number, but it’s true. We also know that it is not possible to write √2 or π as ratios of whole numbers. You cannot trisect an angle, square a circle, or duplicate a cube using only a compass and unmarked straightedge, and neither can anyone else, ever. Euclid’s 5th postulate cannot be proven from the first four. These are all well-known, ancient impossibilities, some of which took more than 2000 years to be understood.

Sometimes in math, a thing that seems impossible turns out to be anything but. Once transcended, imagined impossibilities lead to new advances again and again. Two examples are right under our noses: minus one and its square root. Negative and imaginary were for a long time impossible fictions, total nonsense, but today they are part of the standard numerical toolkit we all take for granted.

The perspective of the impossible gives us access to some of the biggest moments in the history of mathematics: Pierre de Fermat in a few short scribbles described an impossibility of arithmetic that inspired new thinking for more than 350 years before it was finally laid to rest. Evariste Galois showed that the quintic is unsolvable - there is no general formula to solve equations beginning with x^5. Georg Cantor showed that it is impossible to count all the real numbers between 0 and 1, even if you could count forever. Kurt Gödel proved that it is impossible to create a system complicated enough to do basic arithmetic that can also prove its own consistency (there are no inherent contradictions within the rules of the system) or its completeness (answer all of its valid questions), showing that the dream of founding math securely on logic is necessarily doomed.

In math, not only do we transact continually with the impossible, but it is in fact a muse of the highest order. Our modern understandings of form, number, and even the universe owe much to the famous impossible problems above and more.

In this class we will uncover the power of the impossible by taking a historical perspective. We will visit impossibilities throughout the history of mathematics, take them apart, and map their influences. We will also explore the impossible as it can be seen in the mathematics today. By learning how to deal with the impossible, we’ll get a unique inside look at what math is all about. Students will get a chance not just to see important ideas and do problems, but to dig into the contexts that give these mathematical developments meaning.


Yearning for the Impossible by John Stillwell
Mathematics and its History by John Stillwell
Users as Agents of Technological Change by Kline and Pinch
The Structure of SCientific Revolutions by Thomas Kuhn
Things that Make Us Smart by Donald Norman
Other online resources





There are no prerequisites, but math is not a spectator sport; we will be getting our hands dirty. As Euclid supposedly told King Ptolemy, “There is no royal road to mathematics.” Frustration is the name of the game, after all we’re talking about the impossible. It’s going to make your head hurt. In a good way.

We will solve and create problems. So you will need a pencil and paper. You will also need a wastebasket. We will use 21st century tools like computer algebra systems and the typesetting language Latex to do and express math in print and electronically. Students will work in small groups to present two episodes from the history of mathematical impossibility, and write about a third, individually, in research papers. There will also be take-home problem sets.

Students will also be asked to do additional research into the mathematical topics and historical context and use this research to write short essays. A particular focus will likely be analogies between the development of math and sciences and technologies.



Chris Holden is a mathematician for the people. He received his Ph.D. in number theory from the University of Wisconsin-Madison. Originally from Albuquerque, his current research focuses on place-based mobile game design and implementation. Chris enjoys videogames like DDR and Katamari Damacy, and he takes a whole lot of photos.