Suppose you have a collection of pairs of shoes and pairs of matching socks. Let’s think about two questions: Can you create a set containing one shoe from each pair? The answer is yes, for example you could take the left shoe from each pair. Can you create a set containing one sock from each…
Tag: Math
The pigeonhole principle
Let me tell you about one of my favorite things. The pigeon and dove family, Columbidae, is a group of widespread and common birds. Members of this family were domesticated thousands of years ago and they have had symbolic importance throughout history. Many species continue to live closely with humans. People who keep pigeons are…
Let’s talk about the singularity
The concept of the technological singularity is a confused one. Here’s the thing about “general AI” or “self-aware AI”—it will be indistinguishable from really good mimicry of self-aware intelligence. When the singularity happens, we won’t notice. This is an excerpt of a YouTube comment on a video about AI technology. I thought about responding to…
The barber shop pole illusion explained with math
This is a well known optical illusion in which rotation appears to be vertical motion. There are several aspects of the barber pole design which make the illusion more effective, including having multiple stripes, making the stripes different colors, and using rotation. We will strip these things away in order to see what is happening…
Pedagogy recapitulates history (sort of)
Because of the way mathematical ideas build on other mathematical ideas, the order in which these ideas were discovered/invented is often the same as the order in which they are taught. All humans typically start where humanity (is thought to have) started, with counting objects and identifying simple shapes. From there, we begin making calculations….
Operations
You’re certainly already familiar with several mathematical operations, such as addition, subtraction, multiplication, and division. You may also be familiar with the logical operations AND, OR, NOT, XOR, and IF…THEN. More generally, an operation can be defined on any nonempty set. Operations have the following properties. Arithmetic operations We’ve already discussed a few of the…
Metric space
We take for granted that we can measure distances. Most people take for granted that we can represent physical space mathematically and calculate distances between pairs of points. We generally think of ourselves as living in R3, or 3D Euclidean space. Since this is how we represent physical space, it’s generally the most useful (though…
Stories of math notation (part 2): The worst notation in mathematics
Sometimes notation is confusing. One way this can happen is when the same notation is used for multiple different concepts. Usually, if the same notation is used for different things, they occur in different contexts which allows them to be distinguished. In at least one case, the same notation is used for different things in…
Stories of math notation (part 1): y should come before x
Or rather, perhaps both the symbols and positions for x and y should be switched. The convention of writing coordinates as (x, y) is so deeply ingrained that, in my opinion, it causes confusion in situations where x doesn’t naturally come first. Reasons for x first As I see it, the primary reason for writing…
A different view of the multiplication and division algorithms
The “traditional” methods for multiplying or dividing numbers have generally been taught by rote memorization or with minimal explanation. By reframing these processes, I hope to make it more clear what is happening. Column method of multiplication This is similar to the column methods of addition and subtraction, although multiplication is slightly more complicated. Here’s…
thoughts on mind 