One of the nice things about the modern world is that arguing over matters of fact has become less of a thing. With the internet at your fingertips, it is possible to look up a lot of things in a flash. On the other hand, it is still possible to argue motive and interpretation and we are still having a lot of arguments. In this post Occupy Math looks into a type of stalemated argument that cannot be influenced by logic or fact. This type of argument is closely connected to a primary foundation of mathematics. One nice thing — once you notice the wrinkle that shows why some arguments cannot be resolved, you get a tool for understanding that the other person may not be an idiot. The problem is, rather, that their assumptions are different from yours.

# Math Sets You Free

# Weighted Averages – a Useful Tool

We all know what the *average* of a bunch of numbers is: you add up the numbers and divide the total by the number of numbers you added up and you get something in the middle. This sort of average assumes that all the numbers you are adding up are equally important. There are many situations where the numbers are not equally important. This is why we have weighted averages — a weight is another number that says how important a number is in the group of values you are averaging.

The simplest example is when the weights are positive and add to one. This is something that often happens with grades. The teacher might say that “The final grade is 50% of your homework grade plus 30% of your quiz grade plus 20% of your final exam grade”. In this case the weights are 0.5, 0.3, and 0.2. If a student had 86% on homework, 72% on quizzes, and a 91% on their final then their grade would be 0.5×86%+0.3× 72%+0.2× 91%=82.8% for their final grade.

# Mathematical Thinking and the Law

This Occupy Math is partly based on a More Perfect podcast entitled *One Nation, Under Money*. The *More Perfect* series tries to understand and explain US Constitutional law and this podcast is a doozy that looks at a very mathematical approach to the *commerce clause*:

The United States Congress shall have power “To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes.”

– Article 1, Section 8, Clause 3, United States Constitution

The application of mathematical thinking to this clause increased the power of the clause far beyond what someone reading it for the first time might think it had. Consider this while you’re reading the rest of the post: is the interpretation of the clause that arises anything like what the people that wrote the clause thought of while they were writing it?

# Blockchains and Mathematical Freedom

Bitcoin and other crypto-currencies are a type of money secured by the fact that finding them requires an incredible amount of computer power. Letting people use and trust this security requires a very special piece of mathematics that lets a structure called a blockchain make it so that anyone can check the origin and transfer of ownership of each and every piece of cyber-money in the system. The special math is a trapdoor function In today’s post, Occupy Math wants to talk about some things *besides* setting up a crypto-currency you can do with blockchains.

# Trump’s Zero Sum Delusion

This week we examine cooperation and conflict again, revisiting a topic from an earlier Occupy Math entitled “It’s not them or us“. President Donald Trump has recently floated the idea of rejoining the Trans-Pacific Partnership. This was a free trade treaty that the President destroyed — after years of negotiations — just after he took office. The pact was enormously advantageous for the United States: it opened agricultural markets in Japan to US products, for example. Europe stepped up and inked a deal with Japan to supply those food-stuffs, costing American farmers billions. These are, of course, people that voted for the President disproportionately. Largely due to Japan working on it overtime, the remaining countries in the proposed partnership negotiated a version of the agreement (without the US) which was recently signed.

# Corruption: a Mathematical Perspective

Occupy Math will start with a quote from the American Declaration of Independence:

We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed.

This part of the Declaration is saying that the reason we have rules (laws) and government institutions to enforce them is so the rights of all citizens will be respected by all other citizens. Corruption is any of a large number of practices that set aside protection of the rights of individuals to create an advantage for those engaging in the corruption. In this post, Occupy Math will make the case that corruption is obviously disadvantageous to a vast majority — possibly to everyone — using logical argumentation, one of the great branches of mathematics.

This post is not as clearly mathematical as most that appear in Occupy Math, so it may be worth noting that mathematics, in spite of all the focus on arithmetic and algebra, is primarily about the creation and manipulation of formal systems of rules. There is a reason that “math major” is one of the two best majors for getting into law school. It also means that corruption is more obvious to the mathematically inclined. With that excuse, let us continue on to the corruption!

# Fancy Math Talk for Move Half Way

This post is on a really easy method to solve some hard problems, including locating a pretty fractal. It is called the bisection method, but you may know it as a high-low game. One person thinks of a number and the other guesses. The person who knows the number replies “high” or “low”. A good strategy is for the guesser to move half way between their most recent high and low guesses. Bisection is fancy math talk for “move half way”. The thing is that this method can solve hard problems and do **COOL** things. Read on to find out.

# The Cornucopia of Mathematics

Occupy Math is a working researcher and, at least the way he does research, this means that people come to him with problems. In this week’s post we will talk about the type of problems that come up and what the impact of math on the process of solving problems is. The three examples we will use are the design of a wood burning stove, a 3D ultrasound scanner, and a weird abstract computing problem solved with ideas from a science-fiction novel. In all three examples, the point will be that stirring in math to the process helps. The solution became either much simpler or much more useful because math was present and accounted for.

# Why you don’t write a tax plan in a hurry.

Occupy Math has talked before about complex systems. Few human systems are more complex than the economy of a large country like the United States of America. This post reacts to the recent proposed tax bill in the United States — a very complex collection of modifications to a super-complicated system. Since the US economy is pretty much beyond human comprehension, Occupy Math will start with a much simpler complex system. This relatively simple example builds on an earlier post *Evolution Can Do Math That People Can’t*. That post was about evolving cellular automata. The way cellular automata work is that we start with a row of zeros, except the middle three numbers in the row are 1,2,1. The numbers can go from 0 to 7. We sum blocks of five adjacent numbers — getting a sum from 0 to 35 — and look up the new numbers using the table of 36 numbers:

0,2,4,4,0,0,3,0,4,0,6,4,0,2,4,3,0,5,4,3,5,0,3,4,0,5,0,7,7,7,0,0,6,7,6,2

So, for example, if the five numbers in a block sum 15, the new value for the number in the center of the block is “3” because the 15th number (starting counting at zero in the list above) is “3”. We apply the rule over and over to generate rows of new numbers — mapping the numbers to colors to draw a picture. The color map is zero;white, one;red; two;green, three;blue, four;yellow, five;violet; six;cyan; and seven;black. The picture for the rule above is:

Here is a simple experiment in complex systems: change *one number* in the list of numbers that forms the rule for the cellular automata and look at the picture that the new rule draws. There are 36 examples below the fold.

# The Mathematical Value of Diversity

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.