{"id":4,"date":"2016-08-12T14:04:44","date_gmt":"2016-08-12T20:04:44","guid":{"rendered":"http:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/?page_id=4"},"modified":"2016-08-26T10:29:14","modified_gmt":"2016-08-26T16:29:14","slug":"course-overview","status":"publish","type":"page","link":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2016\/","title":{"rendered":"Course Overview (Syllabus)"},"content":{"rendered":"

This syllabus is a work in progress and so far only reflects\u2014my\u2014ideas about the subject matter. The course itself however will depend greatly on the interests and work of those who take it. As they say, no battle plan survives contact with the enemy.\u00a0Expect the history of our work together to turn out differently than what is listed here. To remain organized, we will frequently update this website and stay in touch via Slack. It will not be possible to keep up with what’s going on simply by recalling this plan.<\/em><\/p>\n

About This Course (Course Overview)<\/h2>\n

\u201cYou can\u2019t prove a negative!\u201d 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.<\/p>\n

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\u2019s a little bit harder to show that there is no last prime number, but it\u2019s true. We also know that it is not possible to write (\\sqrt 2)<\/span> or (\\pi)<\/span> 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\u2019s 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.<\/p>\n

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.<\/p>\n

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<\/sup>. Georg Cantor showed that it is impossible to count all the real numbers between 0 and 1, even if you could count forever. Kurt G\u00f6del 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.<\/p>\n

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.<\/p>\n

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\u2019ll get a unique inside look at what math is all about.<\/p>\n

Concepts We Will Use\/Encounter<\/h2>\n

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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

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<\/em>. 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.<\/p>\n

Methods of Learning and Doing<\/h2>\n

Read to learn<\/strong>. Read something over your head ->try to do something related-> work backwards to achieve understanding. 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.<\/p>\n

Make something<\/strong>. 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: \u201cuse it or lose it\u201d 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.<\/p>\n

Requirements<\/h2>\n

It is your job this semester to take on the mantle of mathematician, and help your classmates to do the same. This is simple, like cooking an omelette, but that\u2019s not to say there\u2019s no technique. I\u2019ll try to sum it up:<\/p>\n