“Freedom is the freedom to say that two plus two make four. If that is granted, all else follows.”
–Winston Smith’s journal in George Orwell’s 1984
But is 2+2=4 always true?
Since “post-truth” is the Oxford word of the year, let’s start off this new blog with an Orwellian twist. In 1984, the mathematical expression 2+2=4 is the centerpiece in a battle over Truth. After all, 2+2 obviously equals 4, and the Party’s final and most essential command is for Winston to reject the evidence of his eyes and ears. Winston therefore pleads in his journal, “[T]he obvious, the silly, and the true had got to be defended”; otherwise, the Party can control all of our thoughts.
I am not trying to soften your mind for Big Brother, but … is the mathematical statement 2+2=4 always true? Certainly, 2+2=4 is true for numbers and simple counting: 2 apples plus 2 apples equals 4 apples. However, not everything in the world follows the rules of simple arithmetic.
Take a look at the video below:
Here is the demo as interactive p5js script (move the slider all the way to the right):
It is funny, but sometimes things just don’t add up: 2 rectangles plus 2 rectangles can equal 5 rectangles; 1 piece of clay plus 1 piece of clay can equal 1 piece of clay. And you should really see this.
If you take (or teach) a course on Perception in a university Psychology program, a week or more of the course is typically dedicated to Gestalt psychology. Gestalt psychology is a school of thought based on the idea that the perceptual world can’t be understood from a structural analysis of elemental sensations on their own. Rather, according to Gestalt psychology, we perceive global frameworks that arise from applying simple organizational rules to fundamental elements.
A cliché often used to summarize the thinking behind Gestalt psychology is that the “whole is different than the sum of its parts.” The Winston Smith illusion illustrates this aspect of Gestalt psychology: numbers follow additivity rules, but for our brain, combinations of elements can take on properties that we don’t always expect. The larger gestalt that arises from the actions of simple elements are sometimes referred to as “emergent phenomena.” I will have a lot to say about emergence as the blog continues, but for now I will note that emergence can be mathematically interesting, visually compelling, and instructive about how the brain constructs reality.
So, 2+2 does not always equal 4.
“THERE are three stages in your reintegration,” said O’Brien [to Winston Smith]. “There is learning, there is understanding, and there is acceptance. It is time for you to enter upon the second stage.” 🙂
Some endorsements and credits:
The Winston Smith illusion is based on George Stiny’s clever and mind-expanding book Shape. If you are interested in non-additivity in figures, I strongly recommend his book.
Here is a link:
I also recommend visiting p5js.org. I have been using P5 for my research and my classes. P5 allows me to explore creative coding with my students. P5js is open source and provides great support and easy interactivity; and Dan Schiffman makes some of the best instructional videos on the internet. What’s not to like?
If you haven’t yet read 1984, you really should. If you have read it, you probably should read it again. I recently listened to the book on a recording from Audible:
Image credit for featured image on main page: https://commons.wikimedia.org/wiki/File:USSR_poster_on_first_five-year_plan.png (CC Attribution ShareAlike)