What I’m Reading this Week, Vol. 2

26 Jul 2020

(Similarly Quick) Forward

I set out to write a post like this every Friday, so of course the second week in I had a massive influx of work topped by about 6 hours in Zoom meetings. Oh well! One of the perks of a personal project is that it’s totally under my control: convenient when other responsibilities get in the way.

To try to make up for the short delay, I’m going to challenge myself to write about one of my favorite “Cool Math Things” this week, the Cantor Set. Variety is the spice of life and blogs and all that.

Vaccines stop diseases safely - why all the suspicion?

Joan Donovan

The Nature newsletter is a great source for quick science readings, and for the past few months, of coronavirus updates from scientists (rather than journalists and politicians). This piece is a review of a book by Heidi Larson, Stuck: How Vaccine Rumours start - And Why They Don’t Go Away. The problem of vaccine rejection is a multidisciplinary, community one. Since a vaccine isn’t just for yourself, but has important ramifications for the health of all those around you, public officials have a vested interest in collective vaccination. The observation that having a choice to vaccinate is a matter of not only information and disinformation, but of liberty and dignity, is an excellent nuance on a “debate” that is often the product of inflammatory Facebook posts.

Vaccine hesitancy is a problem of dignity as much as of the abundance of falsehoods: individuals want to have their choices respected, amid growing distrust in authority.

The role that social media plays is fascinating as well. Social media and the internet are both incredibly liberating in that they empower the individual to reach large numbers of people, while allowing users to get content from other users. But, in the opposite vein, social media pushes groupthink in the way that posts are selected to be viewed by algorithms based on popularity. Tack on the particular gimmick of each site (brevity on Twitter, images or video on Instagram, and group associations on Facebook) and each gets their own way to game the system in interesting and potentially harmful ways.

Read the article here.

How Capitalism Drives Cancel Culture

Helen Lewis

Moving past the clickbait-y title for a moment, this article gives a compelling explanation of some of the trends which make up “cancel culture”. Lewis’s “Iron Law of Woke Capitalism”, that “Brands will gravitate toward low-cost, high-noise signals as a substitute for genuine reform, to ensure their survival” definitely appeases the cynic inside all of us. It explains the football team in Washington and the various brands with new, all-black or rainbow logos. While I am a big believer in the fact that language matters - how we communicate impacts ourselves in others - it’s hard not to roll your eyes at a lot of what is done to catch consumer’s attention. I also appreciated her assessment of the popularity of White Fragility a book supposedly about racism which is, in fact, a book about white people, for white people, written by a white women. John McWhorter has a piece in The Atlantic recently where he claims the book condescends on people of color in the way it demands its readers walk on eggshells around topics of racism and social injustice.

The final section, where Lewis disambiguates economic and social radicalism, is a wonderful dichotomy for analysing positions on racism. It reminds me of a question I first encountered in Peggy McIntosh’s seminal essay, White Privilege: Unpacking the Invisible Knapsack: in order to create equality, whether between genders or races or levels of ability, do the dominant groups need to give up some of their power? There are some great conversations to be had about what exactly constitutes “power” and “privilege”. While many of the writers at The Atlantic are progressives, I appreciate the level of nuance given to contentious social debates like this one.

Read the articles by Lewis, McWhorter, and McIntosh at the links on their names.

Something completely different now: The Cantor Set

I promised I’d write a little something about one of my favorite math topics, the Cantor Set. It’s a great example of one of objects of curiousity which I love in math, namely, the concepts of finite and infinite and their interplay. I first heard about the Cantor Set from my advisor, Dr. Peter Thomas, during an office hours discussion. I’m going to walk through the construction of the ternary set, then talk about some of its coolest properties.

Imagine the real number line as you were first taught it in elementary school. We have at the center the number zero, with all positive numbers in increasing order on its right, and all negative numbers in decreasing order on its left. There are are infinite amount of numbers in each direction. Imagine walking along this line in either direction: you’d never stop walking. Alternatively, imagine trying to write down every whole number: 0, 1, 2, 3, 4, 5, 6, …

You’ll never finish.

But, for now, we only care about the numbers between 0 and 1. That includes numbers like 0.25 and 3/4, as well as numbers like half the square root of two and pi over four. These are all equally valid real numbers who lay between 0 and 1 in value.

What is the length of this interval, from 0 to 1? It’s a more complex question than it may seem at first. Often, mathematicians define length as the difference between the furthest apart points: 1 - 0 = 1. This is a sensible measurement system. In the real world, you may say you’re six feet tall, because if you stand next to a measuring tape, the difference between what it reads at the bottom of your foot and its reading at the top of your head is six feet. We also naturally talk about distance as ratios - you may say your commute is a 20 minute car ride, and getting to the park takes about half that time. Therefore, your commute is twice as long as the trip to the park.

What we will aim to do with the interval from 0 to 1 is use this concept of ratios to try to ‘cut out’ all of the numbers in this interval. First, cut the set into thirds. Let $p=[0,1]$, which is a notation for defining $p$ as the set of all numbers between 0 and 1, including 0 and 1. Now, let $p_1 = [0,1/3)$, $p_2 = (1/3, 2/3)$, and $p_3 = (2/3, 1]$ where the parentheses signify excluding the endpoint.

It is fairly obvious that $p_1 + p_2 + p_3 = p$, that is, if we consider all three of these sets $p_i$ we get all of the points that are in $p$.

Now, define a set $q = p_1 + p_3$. That is, $q$ is just $p=[0,1]$ without the middle third. You would agree, of course, that this set is ‘smaller’ than our original set $p$, since all of it’s numbers are in $p$ but not all of $p$ is in $q$. This is a different notion of size than before – now we are interested in how many things are in a set, rather than just the distance between its furthest points. This is perhaps an even more natural way to think about things: we count apples in a basket and say there are 10 apples, or see another basket with 5 pears and say there are more apples than pears, or even that there are twice as many apples as there are pears in the baskets. We’ve simply moved to a more abstract space, where we consider an infinite number of objects at a time. Here, the objects are individual numbers, also called ‘points’ on the real number line. Observe that while there is a clear ‘next step’ in counting discrete objects in real life. If you have 5 apples, the smallest change you can make to that quantity is to add or subtract exactly one apple. You can’t take away a third of apple. However, that’s what we just did with our set of numbers between 0 and 1.

Now, repeat this process of chopping out the middle there ad infinitum. So we go from $[0,1/3) + (2/3, 1)$ to $[0,1/9) + (2/9,3/9) + (6/9,7/9) + (8/9,1]$, and so on, so forth. How many numbers are left?

Well, we certainly still have 0 and 1,, so we have at least two numbers. It can also be shown that we have all numbers which can be expressed of the form $x = c_1/3 + c_2/3^2 + … + c_n/3^n + …$, where $n$ is a natural number (the naturals are all whole numbers 1 and above, the numbers which make sense to count with) and where $c_i$ is either 0 or 2 for all $n$. Here’s an image from Wolfram Mathworld:

Without giving the detailed, analysis-based proofs, that statement above can be shown as well as the fact that it describes an infinite amount of numbers between 0 and 1. The difference between the largest and smallest elements in our final set is still 1, but there are clearly much, much fewer points than started with. Certainly, there are more points which were discarded than there are left. Regardless, we have an infinite number of points remaining – just a “smaller” infinity than those discarded. Of course, we also started with a “larger” infinity of points than is either our final set or all of our discarded points.

This, as it turns out, satisfies a quite a few set descriptions in mathematics. The Cantor Set, as constructed above, is totaly disconnected, perfect, nowhere dense, closed, compact, and of measure zero. Those of you with formal mathematics training are smiling along knowingly, and for the rest of you, well, none of that needs to be meaningful to appreciate the beauty of the set. How can you take infinity away from infinity, and be left with infinity? How do we quantify this set – how large is it, how long is it, how does it compare in various ideas of size with other sets of objects?

It’s interesting, as well, to think about jumping around between the points in the set. What is the next biggest number after 0? Is it 1? No, because $0 < 1/2 < 1$. So is it $1/2$? No, because $0 < 1/4 < 1/2$. Continue ad infinitum, and you begin to recognize a similarity with the construction of the Cantor Set. Namely, there are no two points which are ‘next to each other’ in the final set, since in the process of pruning down the set, we always remove the middle third of any subinterval of points.

In other words, if given a point in $[0,1]$, we could tell you whether or not it is in the Cantor Set by seeing if it can be written with the series expansion above, $x = c_1/3 + c_2/3^2 + …$. However, we could not then write down the next largest number in the set. This describes infinity.

There’s a ton of other cool stuff about the Cantor Set, like how the construction we did (called the Cantor Ternary Set) describes all numbers between 0 and 1 which can be written in ternary without using 1’s. However, what I first found so fascinating about it is the ideas of infinity encapsulated within. That fundamental idea of always being able to find a closer value, even in a set which seems to be so small, is fascinating and beautiful.