“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 (\sqrt 2) or (\pi) 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 x5. 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. We will visit impossibilities throughout the history of mathematics, take them apart, and map their influences. By learning how to deal with the impossible, we’ll get a unique inside look at what math is all about.
You will read math, learn how to research, write, and share mathematical ideas, solve problems. We will largely follow John Stillwell’s excellent Yearning for the Impossible to jump start our journeys, with a little help from Alberto Martinez’ The Cult of Pythagoras and The Mathematical Experience by Philip Davis and Reuben Hersh. These are not our bibles, but our springboards, a way to get used to a more natural mathematical discourse to prepare us for our own research, writing and speaking.
These books will show us another way to look at math besides
- Stuff you’re supposed to cram
- Fluffy, fun, but shallow pop math
We will use them as a way to flex our own intellectual muscles, then work those muscles out by finding our own questions, doing research, and sharing what we find in an as effective as possible form.
Concepts We Will Use/Encounter
Basically, this course is an opportunity to look at mathematics and its development from the perspective of the impossible. This lens will help us see how math is different from other disciplines and bodies of knowledge, and gain new insight into what makes it tick.
The lens of impossibility works in several ways. At once we see that math stands out as a place where you can prove the impossible. Strangely, this lets us guarantee knowledge about infinity (when the finite is impossible). The apparently impossible has a way of melting away into new truths, surprising for how obvious they seem after the fact. And as we discover truly impassible obstacles, we learn so much about everything else.
This path along the impossible is one way to see math not as something immutable but rather something that has come to be over time in often strange ways through the actions and interactions of many people. When it comes to really understanding math, not just plugging and chugging, context matters. Knowing a bit about how science and technology develop over time and how they participate in and are shaped by human affairs can provide some tools to understand math, which is harder to grasp as a kind of thing in the world.
Because we take seriously the idea that math is social process, emergent from and representative of communication among people, we take seriously too the idea that you are apprentices in this discipline. This means you need to not only learn how to see the answer to problems you are assigned but to develop the abilities necessary to communicate mathematics to others. And in this day and age, the textbook is losing its relevance as a product, and pen and paper are losing their relevance as means. You will need to learn how to write mathematics in a way that is web-native, combine analytic thought and computational skill, and find forums and formats for collective mathematical thinking.
Methods of Learning and Doing
Read to learn. Read something over your head ->try to do something related-> work backwards to achieve understanding. You cannot learn math unless you do it. Repetition is the norm. True learning is the building of intuition, the evolution of these concepts from alien externalities to become just another part of your automatic perceptual abilities. This is true of the mathematics we will learn as well as thinking through the related themes we encounter.
Ask Questions. Here, the questions will not all be given to you. You will have to put yourself into the material and ask “why” or “what else”? you will need to look for connections, and go out to find places where there are other people trying to help share the light. These questions can come from our shared reading or your own experiences. You can work on them individually, in class, or through Slack. Actually wanting to know more is the first and most important step to true mathematical learning (or any other subject for that matter).
Make something. Every experience in the name of learning (listening to a lecture, reading a book, watching a show, etc.) should be followed by a reflective task to help the ideas gain weight and grow into those you already have. This principle of learning goes by many names: “use it or lose it” might be the pithiest. So in addition to regular readings, we will have regular bits of creative work to do: response papers, math problems, research, teaching, etc. to accomplish reflection. Doing this regularly is as important as coming to class.
Extension and Revision. The final key aspect to our learning will be finding ways to value it after we have begun. We will go back to things we learned and did in the first weeks, or maybe your youth. We will revise problems for greater exposition, and to become general articles up on the web. We will look back at what we thought we learned and see it as the first step of a long journey.
Growing together. Math is usually pitched as an individual sport. But, besides the way schools find it convenient to sort students based on their performance in standardized mathematical problem solving, the solitary face of math is truly a facade. It truly only makes any sense or gets anywhere as a shared journey. Math only exists as a social phenomenon. It’s just not obvious immediately since the social bits are beneath the surface and much is to be gained by having people believe that their ability to perform on those standardized exams is related to the objective universe and not capricious realities of the social order. Anyway, we will reverse this injustice and do math together. We need to become good listeners, care for more than rushing to the answer, and see that growth in understanding is never very useful if it only happens to one person. The way each individual will do best here will be by doing the most for the learning of our entire group.
