For Quasi-Enthusiasts - Three structures in Math

An Attempt to Explain Mathematical structures to outsiders

Summary

Figure Source: Keenan Crane

In this blog, we give several examples that can help illustrate mathematical structures in daily life.

Mathematics is hard for people who are not aware of its subtlety. One of the reason is that most of mathematical objects encountered in the daily life have too many good structures. A common question addressed by the outsiders is that “why you study this?”, “what is the purpose to study this?” and “why is this important?”

People may not be conscious of the goodness of something until they lost it.

Just like you don’t feel your breath and how important it is at most of the time. Or you don’t really feel how fortune it is to be health until you get illness.

In my point of view, to explain mathematics to the outsiders, it is important to illustrate the difficulty when the good properties are abandoned.

Here are three simple mathematical structures of abstract objects:

  1. Order
  2. Transformation
  3. Metric

which roughly correspond to the three main area in mathematics:

  1. Analysis
  2. Algebra
  3. Geometry

Numbers are the most typical mathematics object which contains all these three structures in a very intuitive way. By Peano’s Axioms, natural numbers are ordered in the sense of other numbers’ successor. And this order structure is extended to the real number.

Order, a target of analysis

Order structure are mainly studied by the order theory. There are two useful concepts total order, partial order in the set theory. Order structure are investigated by using binary relations. The inequality $a<b$ is an intuitive examples. Also, $Z\subset R$ is another example.

Examples

  1. Alice is older than Bob, Charles is older than Bob, but we don’t know information about the height between Alice and Charles.
  2. Your friend went a new restaurant and told you the food is good. How do you get the information about how would you feel about that restaurant?
  3. Consider the following concepts: big vs. small, red vs. blue. Why it isn’t straight forward to compare two colors?
  4. When you say sun is brighter than a light bulb, what are you really trying to address?

Transformation, a goal of algebra

Transformation structure studies what the outcome when apply an operation to an object. Basically, it studies the statement like $x\in M, y\in N, f: M \rightarrow N, f(x)=y$.

Examples

  1. Group Example: rubric cube
  2. Dynamics Example: Car moving, Earth orbit
  3. Storage in Computer: a finite field
  4. Switch of light, switch on make the light to the state on.

Distance, a playground of geometry

Determine the distance between objects. The order or transformation structure don’t contain the information about the geometrical relation between points.

Examples

Rotation of a ball is a ball

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title: "An Attempt to Explain Mathematical structures to outsiders"
date: 2022-10-20
img: posts/20221020/post7-header.webp
tags: [Mathematics, Philosophy of Mathematics, Popular]
category: theory
author: Hanchun Wang
description: "Order, Transformation, Geometry"
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---