Mathematics: Art of Seeing

Hi everyone! How’ve you been this time around? It took me a bit while to actually come out with an idea on what should be on my first math post. Without further ado, let’s get started and hope you enjoy reading this.

Today, I would like to bring you guys to one part of the maths universe that most people wouldn’t think it is mathematics. As the title suggests, I am going to introduce a mathematical concept that teaches how to see (a collection of) things in many different ways.

Motivation

Say that we have a collection of objects in our sights-it can be anything like a group of people or a bunch of candies. From this collection of objects, you know that you need only one of them. For instance, you feel like watching a movie at the midnight and you opened Netflix app on your smart device. Once you open the app, you will immediately see loads of Netflix shows. How would you find/decide which one to watch that night? Let’s consider two possibilities.

Case 1

You probably don’t have any specific title, but you have a preference on which genre you are interested in. Thanks to the Netflix algorithm and team, the shows have been sorted out by their genres. With this help, you were able to narrow down your search to the genre of interest.

Case 2

You probably have a specific title in your mind so you go straight to the search engine and spell it out. Every time you inserted a letter, a list of possible shows was displayed below the search engine.

In Case 1, what you did is you “see” the shows according to their genres. Meanwhile, in Case 2, you “see” the shows by their title/name. In any of the cases, you allow yourselves to “see” a smaller collection of objects, which is a natural thing to do. Then, the question is, how would you choose to “see”? In particular, how would you group the objects?

How to “see”

What I will give you here is only one art of seeing things. They are a few other ways and I will cover them in other posts if time permits.

Given any collection of objects, you can sort them into categories so that:

1. Combination of categories is also considered as a category
2. The intersection of two or more categories forms a new category

Let’s jump to our primary example, Netflix.

Case 1: Categorized by genres

If we have comedy and action as our genres, then we have two new categories induced by the two categories. The first one is the category of “either comedy or action” which includes all shows in the action genre and all shows in the comedy genre. The second category is the category of shows which are considered as both comedy and action. When you are a bit indecisive about which two genre is more preferable, look into the first category. When you want the touch of both two genres in a single show, then go for the second category.

Case 2: Categorized by name/title

Let’s say we have the categories of shows whose title contains the letter “A” and “T”. We then have a new category of shows whose title contains either “A” or “T”. Also, we have a new category of shows whose title contains both “A” and “T”. If you are neutral watching either Aladdin or Thor, then the first category suits you. If you are determined to watch Attack on Titan, the latter category will come in handy.

Summary

From a big collection of things, you are allowed to “see” things in a few different ways. Different ways potentially help you achieve different goals/objectives. This is not exactly how mathematicians see things. It is more like the natural way people would do when looking at a large group of objects. There is more to this art of seeing that mathematicians have been studying. For instance, we study different ways of seeing the collection of real numbers ($1$, $1000$, e, $\pi$, $\frac{5}{2}$, $\sqrt{2}$). From this study, we are able to develop more theories on properties for each way of seeing.

Exercise

• Could describe different ways of seeing:
  • when you search for songs in the Spotify app and
  • also when you are looking for a drink at a store
• For mathematicians, could you tell which mathematical concepts are relevant?