Although this course is intended to open up as you find your interests, I don\u2019t want us to worry about getting lost. I have arranged a sequence of themes, or modules, that take us through some interesting ideas in the history of the impossible. There are more modules below than we will have time for, so to some extent, what we pursue will match the expressed needs of the class.\u00a0<\/p>\n\n\n\n
Roughly, I expect to spend about three class periods on each module. That leaves us some room to go over and to include class time for learning about writing, math software, the development of science and technology, math culture, etc.<\/p>\n\n\n\n
Each module has a primary reading to get us started. A first pass of this reading, and some basic notes for further inquiry is a basic expectation from which our class discussion of the material and further work will depart.<\/p>\n\n\n\n
Here, I give an overview of each module. For those we take on, I will also post a more detailed page as a resource to help us dig in.<\/p>\n\n\n\n
Chapter 1 in Yearning for the Impossible<\/em>. What are numbers? We begin to look at this deceptively simple question with the first group “known” to have wrestled with it, the Pythagoreans. Their religious vision of a universe based on numbers ran into some problems when confronting geometry and music. As they say, the devil is in the details. We will see how impossible feats for numbers eventually resulted in an enlargement of the concept of number itself, and connect this arcana of pure math to the construction of musical scales.<\/p>\n\n\n\n Since this module will also be our introduction to the discipline of mathematics, we will spend a bit of time introducing and examining the peculiar values, ideologies, epistemologies, and methods used by mathematicians: what they care about and what they consider their work to be.<\/p>\n\n\n\n Check out the resource page<\/a> for this module!<\/p>\n\n\n\n After we dive into the math a bit and make sure everyone is interested in doing that for a whole semester (no one needs to take this class\u2014drop if this is not interesting enough to keep up with), we\u2019ll get set up with our blogs and learning how to Latex.<\/p>\n\n\n\n All those little symbols and diagrams you see in math books are not on a standard keyboard. Mathematicians do not have different keyboards. They produce text using a typesetting language called Latex (pronounced la-tech) developed by Donald Knuth and others starting in 1978. There\u2019s an interesting story here and some skills to pick up. Most mathematicians and scientists use Latex to write their articles and talks. This will be our introduction to using the Latex language to produce nice looking math on a computer.<\/p>\n\n\n\n The other angle on Latex that has interest for us is the way in which it is a working tool of these professions, common to many, and requiring skills to use, yet there is little official mention of it in any of the education of young people working their ways into these professions. It is a good example of something that is implicit knowledge within a discipline. Implicit knowledge is just as important to have as explicit knowledge, but because there is not a class for it or mention of it in a syllabus, you might wonder when and where a student is to pick it up or to even know that they need to know it. Despite the fact that it is used almost universally, once you start googling, you will find that the help and directions out there are nowhere near as complete or informative as you might like. Try comparing it to something else that is technically very similar like another typesetting language, say Markdown.<\/p>\n\n\n\n As nice as Latex is for typesetting math, it is designed with the idea of using a computer to produce ink on paper. There are some inherent differences between this and a language like html, where the final destination is a screen. We will look at some of the historical attempts to find a way to get math looking right on the web and a popular recent contender, a javascript library called MathJax.<\/p>\n\n\n\n This will also be our introduction to talking about writing on the web. We will look at some of the platforms and methods available. We will consider what kinds of mathematical writing is already out there on the web and where you can and are able to contribute.<\/p>\n\n\n\n Chapter 1 in Roads to Infinity<\/strong>. The Pythagoreans and the greeks who followed them tried to stay away from infinity. In a basic sense, it is a difficult thing to treat the idea of something going on forever as an object of reason. Not only does it make your head hurt but infinity quickly leads to paradoxes. You may have been chided by a math teacher to stay away from it because \u201cinfinity is not a number!\u201d And this is pretty much the main line of thinking on infinity for the last 2000 years. In the 1600\u2019s, especially due to the leading influence of Issac Newton, mathematicians began to use infinity again, but they were still scared of it, usually careful only to refer to potential<\/em> infinities. But in 1847, Georg Cantor put a stake in the ground. He totally changed the way we think about infinity, and in doing so came up with the language that all modern math is written in, the language of sets.<\/p>\n\n\n\n We will learn how to count infinity.<\/p>\n\n\n\n One of the goals of building academic knowledge is not to simply change what\u2019s in your head but to pave the way for those who follow you. Sometimes the best part of a book is not what is written but who is cited, the reading you do next. The time-honored annotated bibliography is this when it is at its best. But the internet has developed its own forms of memory aids. Social bookmarking sites like Delicious make it possible to keep track of and organize knowledge that is scattered over vast terrain. Pinterest is similar but with a strong visual and social aspect reflecting the concept of sharing knowledge rather than squirreling it away. There are also academically minded programs like the excellent open-source Zotero that try to combine the best of both worlds.<\/p>\n\n\n\n In this class, we want to keep track of the interesting writing on the web so that future classes might start a leg up and so that interested parties with similar aims, wherever they may be, can benefit from the research done by us. So we will play with various bookmarking and citation strategies and adopt at least one to record our paths this semester.<\/p>\n\n\n\n Chapter 2 in Yearning<\/strong>. Today we take negative numbers for granted, and expect even small children to see them as a basic way to talk about and understand the world. But it was not so long ago that they were seen as impossible nonsense. It wasn\u2019t until 1500\u20131800 somewhere that they became part of the mathematical canon in enlightenment Europe. How could something so basic today seem impossible for so long?<\/p>\n\n\n\n Around the same time negatives became Kosher, so did another kind of number: Complex numbers. Imaginary, complex – our words for them suggest they are obtuse, strange creatures. Maybe as a result, most people today don\u2019t seem to think they\u2019re quite real. This is an injustice. To mathematicians, engineers, scientists, etc. Complex numbers are not weird at all. They make the world make sense. There is nothing more concrete or real.<\/p>\n\n\n\n We will see some of the cool tricks that Complex numbers can do, and by comparing their history to that of negative numbers, gain some perspective on the invention and acceptance of mathematical ideas.<\/p>\n\n\n\n Users as Agents of Technological Change: The Social Construction of the Automobile in the Rural United States<\/strong> by Ronald Kline and Trevor Pinch. People mostly have a pretty naive assumption about how science and technology develop through time. We suppose that science develops logically, through the scientific method, a sort of natural selection of ideas against the truth of the world. We also assume that technologies win users and persist through a survival of the fittest. In redeeming ourselves, we need to see that \u201cbetter\u201d is a slippery concept itself, and that it is not always responsible for the persistence of old ideas or the success of new ones.<\/p>\n\n\n\n We have already seen enough mathematical developments that challenge the naive model. To gain tools to start asking good questions about how mathematical ideas change over time, we will look at some thinking about how science and technology develop. Part of this is making the analogy of math as science and math as technology. These analogies themselves require interrogation and invite others.<\/p>\n\n\n\n Chapter 4 in Yearning<\/strong>. Previously, we mentioned that European mathematicians of the enlightenment began to use infinity. Newton et al got a little carried away by the power of these methods, and never really confronted the fact that what they were doing, in the strictest sense, didn\u2019t make sense. Calculus worked on the basis of infinitesimals, (dx)\u2019s and (dy)\u2019s. Bishop George Berkeley, in a scathing indictment, called them \u201cthe ghosts of departed quantities.\u201d In a nutshell, here is the problem. Sometimes we treat (dx) like a number. We divide other quantities by it. Other times we take it to be so small it is (0). (2x + dx = 2x). How could something both be zero and not zero at the same time?<\/p>\n\n\n\n In the next couple hundred years, mathematicians found a way around the contradictory infinitesimals. Essentially, this is the bureaucratic stuff about limits you have to be careful with on Calculus tests. We will see how the fixes work, but we will mostly be interested in how something that is nonsense could nevertheless be a powerful tool. We will also see how, more recently, Abraham Robinson vindicated infinitesimals by creating a new number system called the hyperreals.<\/p>\n\n\n\n Chapter 3 in Yearning<\/strong>. There is almost nothing so sacred as our intuition about parallel lines. It was most famously encoded by Euclid, in the Elements, as one of five basic assumptions about geometry, and this geometry was long taken as the foundation for cosmology. It wasn\u2019t until more than 2000 years later, around 1850, that anyone was bold and insightful enough to discard it to build a different geometry.<\/p>\n\n\n\n However, this is not entirely true. Perspective drawing, popular through the Renaissance, encodes a geometric system where parallel lines do in fact meet. Projective geometry as it is called is in fact a perfect way to look at where these alternate universes might be and what they feel like. We will both look at some basics of projective geometry and spend some time getting to know the axiomatic method, whereby mathematicians build systems of thought like geometry and algebra.<\/p>\n\n\n\n Chapter 5 in Yearning<\/strong>. Is there anything more fundamental to how we think of the world than the basic shape of space? From Euclid\u2019s The Elements until around 1850, space other than flat space was impossible, total nonsense. Unthinkable even. Beginning from our previous discovery of alternatives to Euclid\u2019s parallel postulate, we will get to know curved space.<\/p>\n\n\n\n Intro and Chapter 1 in The Structure of Scientific Revolutions<\/strong> by Thomas Kuhn. Similar to how we saw the automobile evolve in response to its interpretation by people in rural america, Kuhn asks us to look at the history of science as revolving around social contingency, not the inherent correctness of ideas. He aims to correct the usual textbook revisionist history where a picture of logical determinism is painted. Just as it might not make sense to say who discovered Oxygen because just what was discovered only makes sense from the framework of a modern understanding of chemical elements, we might be able to gain new perspectives on the revolutions in math that we have now witnessed.<\/p>\n\n\n\n Chapter 6 in Yearning<\/strong>. It is impossible to imagine a fourth perpendicular to the three dimensions we typically consider. Coordinates makes it easy today. How did we get there? We will take a look at how Hamilton tried to devise a system of numbers beyond the 2-D complex numbers, and look at how algebraic concepts has allowed mathematicians to take the idea of dimension from something concrete and simple to something that they can pull out of their pockets without a second thought.<\/p>\n\n\n\n Chapter 7 in Yearning<\/strong>. Primes are indivisible. Another basic truth of grade school arithmetic. Another impossibility successfully challenged in the 19th Century. Another instance where this challenge created brand new worlds to explore.<\/p>\n\n\n\n We will explore some of these new worlds, see new primes, and use new numbers. We will see how arithmetic itself breaks down and how, through the creation of another magic entity, the ideal, it can be built up again.<\/p>\n\n\n\n What is SageMathCloud?<\/a> A blog post by Sage and SMC creator William Stein<\/p>\n","protected":false},"excerpt":{"rendered":" Although this course is intended to open up as you find your interests, I don\u2019t want us to worry about getting lost. I have arranged a sequence of themes, or modules, that take us through some interesting ideas in the history of the impossible. There are more modules below than we will have time for, … Continue reading “Content Overview”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":1,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-119","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/PbpbST-1V","_links":{"self":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/119","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/comments?post=119"}],"version-history":[{"count":2,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/119\/revisions"}],"predecessor-version":[{"id":122,"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/pages\/119\/revisions\/122"}],"wp:attachment":[{"href":"https:\/\/mathematicalimpossibilities.localgameslab.org\/2018\/wp-json\/wp\/v2\/media?parent=119"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Writing Math on the Web<\/h2>\n\n\n\n
$LaTeX$: Writing Mathematics Electronically<\/h3>\n\n\n\n
Resources<\/h4>\n\n\n\n
MathJax: Writing Mathematics on the Web<\/h3>\n\n\n\n
Infinity Is Not a Number<\/h2>\n\n\n\n
Keeping Track of It All<\/h2>\n\n\n\n
You Can\u2019t Take 7 Away From 5 or the Square Root of \u20131<\/h2>\n\n\n\n
The Social Construction of Technology<\/h2>\n\n\n\n
Infinitesimals: The Ghosts of Departed Quantities<\/h2>\n\n\n\n
Parallel Lines Do Not Meet<\/h2>\n\n\n\n
Space Is Flat!<\/h2>\n\n\n\n
Paradigms and Revolutions<\/h2>\n\n\n\n
There Is No Such Thing as the 4th Dimension<\/h2>\n\n\n\n
You Can\u2019t Divide Into a Prime Number<\/h2>\n\n\n\n
Math and Computers<\/h2>\n\n\n\n