In this post, Occupy Math tackles the thorny issue of diversity from mathematical perspective. Many of us take it as given that diversity is a good thing — while some of us think it is a plot to destroy our culture — but there is a mathematical perspective from which diversity is intrinsically valuable. When a crisis comes, it is difficult to predict what qualities are needed to deal with it. This means that a diverse population hedges its bets while a homogeneous society is placing an all-or-nothing bet on their survival. This is why the genetic diversity of an endangered species is of such concern to conservation, but similar principles apply to everything from problem solving to human society. This even shows up in our corporate culture where believing in a “one true way” of thinking about or solving problems can be a barrier to progress.
There is an inaccurate stereotype of millennials as lazy, entitled, self-absorbed lumps who live with their parents. When Occupy Math’s editor graduated from Bard College, it took her a couple of internships and several years to find a good position. Her grandmother could not understand why finding a job took so long and was concerned that lack of zeal might be the problem. What’s actually going on is that good jobs are far rarer than they used to be and affordable housing is also really hard to find, hence living with their parents. This inaccurate stereotype is an example of a smooshing error. This is an error where you combine a whole bunch of diverse factors (job availability, housing prices, the impact of automation and the internet) into one thing in your head and then map it onto your own experience, drawing a wildly incorrect conclusion. In the late 1940s (when the above-mentioned grandmother was hunting for a job), inability to find a position probably would have required laziness. It was a different world then.
In his awesome Discworld Series, Terry Pratchett often included twisted, hysterical versions of modern science and scientific myths. One of my favorites among these is the quantum weather butterfly. In this post we are going to talk about the famous butterfly effect and the odd notion of deterministic chaos as well as one of the biggest human misunderstandings of an obvious mathematical truth.
When people figure out how bad something is, they usually try and figure out how much changing one thing changes another and then use that to calculate the estimated impact. If you raise the price of a car model by $500 and sales drop 0.5% then you estimate a $1,000 dollar increase would drop sales about 1%. If several things are changing, you check the impact of each one individually and add up the results. In Occupy Math’s calculus+physics class we actually work out (as part of error analysis in physics) when this type of estimate works. The answer is when the factors either don’t interact or when the changes in all factors are very small. Most changes are pretty small so that means this “calculate them individually” approach works pretty well. In this week’s post we look at situations where this method of estimating impact fails catastrophically.
This week’s post takes a look at the most usual way that you can become a professional mathematician: going to graduate school. The first university degree in mathematics gets you ready for a lot, working for a bank, starting up the ladder of the actuarial profession, or becoming a codebreaker for the military. Because the advanced reasoning skills are useful, you can usually find employment as a system administrator or as a manager of some sort. Someone who has been trained in mathematics gets an edge in many careers. This post is about moving on to the level where your job is to invent new mathematics.
Humans have a natural ability to do logic predicated on learning rules from their environment. Training in mathematics improves your logic — and makes you more effective in an argument (at least the polite sort). It’s often humorous when somebody does not follow those rules. In the show Parks and Recreation, actress Amy Poehler plays the director of the Parks and Recreation department in a small town in Indiana. At one point a citizen, at a town meeting, says this. “I found a sandwich on a bench in one of your parks! (pause) Why wasn’t there any mayonnaise on it?” Something that sounds like a complaint about littering turns in a completely unexpected direction. Occupy Math starts with this because the fact that this condiment twist in the citizen’s complaint was funny means that there is hope in a moderately awful situation.
This week’s Occupy Math is venturing again into the sociology and politics of education in the service of addressing a pressing problem. There are places where Occupy Math managed to wedge in a little math — and there are tips on self-defense for those of you who might pay for educational opportunities. The basic thesis of this post is that trying to run education like a business degrades education and also fails at the normal goals of business. The basis of the problem is that education is critical, people really want it, which means the demand for it is inelastic (doesn’t change much with price), which in turn means price can get totally out of control. This also makes education a fertile ground for con-artists, who are always willing to exploit people who really want something.
This week in Occupy Math, we proudly announce a book published by Dr. Andrew McEachern, GAME THEORY: A Classical Introduction, Mathematical Games, and the Tournament. Game theory is a formal structure for studying and resolving conflict and encouraging cooperation that rephrases cooperation and conflict as a game. Andrew developed and taught a course in game theory taken by advanced students from many programs while he was at Queens University. This book is a text based on the course he taught and it is part of an effort to bring textbook prices under control. The book introduces the classical analysis used in game theory — his exposition of The Lady or the Tiger is wonderful — but Andrew also introduces material outside of the standard game theory fare. These include the math behind the fraction teaching game that Dr. McEachern and Occupy Math are developing and techniques for designing fair, balanced tournaments for anything from Prisoner’s Dilemma to Basketball. The book is a text for a course for non-majors that nevertheless has a solid mathematical foundation. We now ask Dr. McEachern a few questions about his book.
Occupy Math is going to look at a simple piece of math that is ignored or, worse, abused by researchers in many fields. It amounts to an example of ignorance of statistics that leads to publishing results that are bogus and so impossible to replicate. This problem is called the replication crisis because many important results seem to disappear when other researchers try and reproduce them. Occasionally this is the result of actual fraud — but more often ignorance of simple facts about statistics can let you publish a paper whose results cannot be replicated (because its results are actually wrong) without even noticing you’re doing it. There is also a separate problem — it is very difficult to completely describe an experiment, which means that the people trying to reproduce your results may not be doing quite the same experiment. That last is a big problem, but not what Occupy Math is looking at today.
The core message of today’s post is that peer-reviewed results in a top journal are sometimes wrong because we don’t teach statistics properly.
This week Occupy Math takes a trip to the land of Freedonia which is beset by a vile dragon that menaces its democracy. A small African nation with a diverse population and the magnificent port city of Great Haven, Freedonia is a constitutional democracy modeled on the American experiment, an active, participatory, free democracy and, until recently, with a vibrant and open economy. Founded in the early 20th century by a largely peaceful revolution organized by tribal leaders — advised by the famous explorer and polymath, Captain Spaulding — Freedonia has been synonymous with hope for generations. Recently, however, the economy has been experiencing problems with corruption. Nepotistic awards of government contracts to incompetent nephews and corrupt back-room deals have taken the economy away from the hard-working farmers, shopkeepers, and craftsmen who have been the backbone of Freedonia society. UN monitors certify each of the biannual elections as free and fair but, somehow, in spite of public outrage, the Lucarian party ekes out a bare majority and restores the corrupt Prime Minister, Joseph Cagliostro to power. What dark force is subverting democracy in Freedonia? Let’s ask no lesser authority that the Governator himself!
Gerrymandering is a subtle way of subverting democracy — and the vorpal sword that can slay it is edged with mathematics.