Check Your Assumptions at the Door.

When Occupy Math lived by himself in Pasadena, California, his custom was to have breakfast at Burger Continental on South Lake Avenue and read the LA Times Sunday edition. This makes for a long breakfast on all fronts. One story that caught his eye was about how far a cat could fall without dying. Eight stories was pretty safe, nine-to-twelve stories usually resulted in injury, changing to death at thirteen or more stories. This is an old memory so the numbers may be a bit off – and do not attempt any experimental replication!

The Times got a lot of hate mail about reporting on and encouraging evil (often “government”) scientists. Many of their readers (and more than half the people Occupy Math tells this story to) had a vision of an expressionless android in a lab coat flinging cats out of skyscraper windows. “Subject 34 ‘Fluffy’, 10 stories, pull”, followed by a descending yowl of terror. The study that the Times was reporting was compiled by a vet who, when a client brought in a cat to have it checked after falling, asked from what height it had fallen. People reported what floor they lived on when the cat went out the window after a bird or whatever, which forced the choice of reporting statistic. After compiling the data, the vet published his findings, which are mostly useful for deciding if it is safe for a cat-owner to open a window.

Why did so many people assume that funded government researchers were deliberately throwing cats out the window?

Since the creation of the atom bomb, people have had doubts about scientists and movies have a lot of obsessed, inhuman scientist characters. People also thinks science only happens in labs or at least highly controlled conditions. Controlled conditions are really nice when you can get them, but they are often not available. A lot of what Occupy Math works on is sorting through the debris of carelessly gathered data from the real world. The post with big data comes big responsibility may help you understand the degree to which a lot of research deals with whatever data walks in the door.

Consider the case of astronomy. How on earth would you do a controlled experiment about how a star explodes? Occupy Math hopes we never find out. Astronomers do have experiments, but what they do is observe the sky, collecting huge amounts of data about what is there, and then reason about their observations. What we know about exploding stars comes from observing stars that happened to explode. The unavoidable need to deal with imperfect data is one of the big reasons for statistics. If 93% of your samples are healthy and 7% have the disease you’re studying, the statistical tools you use need to be different. Since accuracy typically increases with sample size, your conclusions about the unwell minority will have larger uncertainty.

It may seem odd for Occupy Math to say this, but math does not exist in the real world. Rather, math sits outside of the real world and its truths transcend all boundaries. How on earth can puny humans use something like that? Simple: we make assumptions, which we call axioms. The big difference between these mathematical assumptions and the assumptions that people make all the time is that

A mathematician knows the assumptions are there.

Let’s look at a few examples. A math question that often gets asked is does infinity exist? There is a large and ongoing debate on this topic, especially about examples of infinity in “reality”. In mathematics the standard list of axioms contains the “Axiom of Infinity” which assumes an infinite set exists. So mathematicians don’t have an official opinion on the existence of infinity, we just (usually) assume it exists.

In fact, long ago, Occupy Math was participating in an online discussion called “sci.math” on Usenet. The infinity debate was in full swing and someone who knew Occupy Math was a math professor asked him to weigh in. Occupy Math said “For mathematicians, infinity is an assumption. We don’t think it exists and we don’t think it doesn’t exist. We can work with either assumption.” This made several people in the debate very angry — people like simple, certain answers, I guess.

Why bother to worry about your assumptions?

The axiom of plane geometry that Euclid was most reluctant to include in his postulates was the parallel postulate. This postulate says that if two lines cross a third line and are getting closer to one another then the lines meet on that side of the third line. It’s a pretty technical statement (Occupy Math softened it) and it looks like it ought to follow from other things. Euclid only included it in his list of necessary assumptions after trying really hard to prove it. What’s cool is this. You can deny the parallel postulate and, as a result, get an entirely different type of geometry: Non-Euclidean geometry

This gives us the motive for worrying about our assumptions: an assumption that you are aware of is one that you can challenge. This is another point where Occupy Math’s slogan “Math is the right of all free people” comes into play. In Caesar and Cleopatra, George Bernard Shaw had Caesar say:

“Pardon him, Theodotus, he is a barbarian, and thinks that the

customs of his tribe and island are the laws of nature.”

This is a wonderful quote (see the play!) for illustrating that people are so tied to their assumptions they do not even see they are there. This causes a lot of trouble in the real world.

Between the 2012 and 2016 elections, many pollsters shifted from reporting the percentage difference in the polls to the probability that candidates would win. This means “Ahead by 4%, 44% to 48%” became “72% chance of winning”. In 2012 having 72% (of the vote) would have been a never-before-seen blowout. In 2016 a 72% chance of winning was razor thin and died. How many people did not notice these two 72% numbers were completely different? There were other unexamined assumptions in the campaign – the assumptions that most voters would be people that voted in the last election or that white women would favor Hillary, for example. In any case, the dominant factor in this election was people that didn’t vote, possibly because they thought their candidate was way ahead due to a change in the way things were reported.

It is critical to know what your assumptions are or they could eat you.

Absolute certainty is an unavailable quantity. You have to make assumptions in order to go on with life. Mathematicians and scientists try hard to make sure their assumptions are ones they are aware of and they are committed to re-examining their assumptions when any evidence at all suggests there might be a problem. In this these disciplines are almost unique as human institutions. The default state of people is to believe what they were taught as children — or its exact opposite during periods of rebellion. Carefully thinking though your assumptions and discarding those that do not measure up is potentially beneficial, but all too rare.

Occupy Math sees that this places a huge burden on teachers to help their students have humane and sensible assumptions. It also shows another reason why teaching as much math as possible is helpful. Occupy Math has noted that different points of view are inestimably valuable in math, but in fact they are valuable across the board. Learning about other times and other cultures will help your math skills by enhancing your diversity of viewpoints, but it will also help you to see your own assumptions when you encounter others that do not share them. Have you seen people having a problem because of assumptions they were not aware of? Comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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