What Are Numbers?

Today I am going to re explain a concept most of you learned before kindergarten. The discussion will go from what numbers are and why we use them to an incredibly verbose discussion of how counting works and why the mechanisms of it are important. Let's hope I know what I am doing. Cue intro.

Webster's dictionary defines number this way:

• a word or symbol (such as “five” or “16”) that represents a specific amount or quantity
• a number or a set of numbers and other symbols that is used to identify a person or thing
• a person who is identified by a number and not treated in a personal or friendly way.

These were the simple definitions, they actually go a bit further.

1. (1) : a sum of units : total (2) : complement 1b (3) : an indefinite usually large total <a number of members were absent> <the number of elderly is rising> (4) plural : a numerous group : many (5): a numerical preponderance
2. (1) : the characteristic of an individual by which it is treated as a unit or of a collection by which it is treated in terms of units (2) : an ascertainable total <bugs beyond number>
3. (1) : a unit belonging to an abstract mathematical system and subject to specified laws of succession, addition, and multiplication; especially : natural number (2) : an element (as π) of any of many mathematical systems obtained by extension of or analogy with the natural number system(3) plural : arithmetic

To be fair, english has a broader range of meaning than mathematics so lets go over the definitions that are most relevant to us. At the end of the course, I want you to understand definition number three above. But we might not be there yet. Let's instead with a simpler definition, a word or symbol (such as “five” or “16”) that represents a specific amount or quantity. Why are we using this definition? Because this is the definition I could see explaining to my 4 year old niece and getting away with it. But there is something funky about it. It's lazy to simply say words like "amount" and "quantity" without giving at least a rough definition of those ideas. Turns out it is harder to do than it looks. Most people learn it through inference like most people learn their early words.

What do these things have in common?

If you guessed the word three, congratulations you have excellent pattern recognition! But what is three? We are talking about a wide variety of completely different objects. Where does this concept that seems to apply to all of these groups come from? Turns out it is all about relationships. The pattern that is recognized as "three" is an example of a type of relationship between groups of things we call a cardinality. A cardinality is the idea that two groups are related if we can pair up the members without any leftover. Let's take the last two groups, the Three Stooges and the Three Amigos.

I can line up the members so a member of one group is associated with the member of another group.

Obviously, we can produce this effect more than one way, but it is enough that one association like this exists. If say I tried to associate the Three Amigos with the letters A ,B ,C , and D or with the colors white and black. This type of association would not be possible. With the letters, all the Amigos could be matched with a letter but there would be one letter left over. If I tried to match them up with the colors one of the amigos would be left out. This process works with any two groups of things. If a map from one group can go to another group where each piece goes to something and there are no repeats, then the two groups have the same number of elements. Oddly enough, this is also how mathematicians define the most basic numbers as we normally think of them.

The system I have described so far has a couple flaws. We can show two sets have the same number but we can't tell yet with just this what the different numbers are. It turns out this is more complicated than it looks. Most of the ways we have to find all the numbers would probably work, because hindsight is wonderful. Most people have worked with numbers for years, so we intrinsically know what they are and we can just name them off, (0, 1, 2, 3, ...). But as anyone who has ever tried to write for a dictionary could tell you, you can't really do that to define something. So to find all the cardinalities, we need two things a way to name them and a way to show the naming method covers all the possibilities. First question is relatively easy. We can name all the numbers in a fairly consistent way. Turns out showing that a method covers all the numbers is harder than it looks. There was a paper called Principia Mathematica by Russell and Whitehead that took 360 pages to solve the complex theorem 1+1=2. As odd as this sounds, problems and contradictions like this are not uncommon in mathematics. We are going to talk more about this when we discuss infinity.

For now, we can say with certainty the concept of cardinality is an effective definition of number and 

that what I am going to talk about next is an effective way of naming them all, even though I can't prove it in this post. Which is where we get into ordinal numbers. Ordinal numbers are a categorization process for numbers as we understand them. In most spoken languages, ordinal numbers represent a literal order( first, second, third,...). Math they are defined as an actual order. an association from a number to the set of numbers that says what is less than that number. Zero has an empty set. One has 0 because the set including just zero. Two has 0 and 1. three has 0,1, and 2 and so on. This order is also exploited in math to define numbers. Quite a few sets can have a particular numerical characteristic, as we have noticed earlier, so mathematicians use ordinal numbers to create example elements of these numerical categories. We can treat these examples as the numbers themselves. Zero is the empty set because that set has no elements. One now is the set with zero in it. Two becomes the set with zero and one. Any group of things with the same number as these sets is now a part of that category and can be easily identified abstractly.

Why would this matter? We usually learn how to count by the time we are 3 or 4. And we even learn to use these ideas without having to define them so specifically. If I ask my niece in kindergarten to determine the cardinality of the pieces of candy she received on Halloween, she would look at me funny and if she were feeling particularly precocious she would ask what "cardily" and "deterine" mean. Or they can do what my niece did and look at you with the deer in the headlights look their choice. But even if she didn't understand the jargon I've seen her use the tools behind it. When presented with a set of things she needs the quantity of she assigns an arbitrary order to that set. Then, she matches that order to a predetermined one say: one, two, three, four,... . And the final member of this set suddenly becomes the quantity. For something so basic, this can seem like overkill. but as the concepts become more advanced. This specificity becomes very useful.

There are several overall themes in mathematics that are helpful to remember here. One, everything is about patterns and relationships and numbers are no different. Two, there are different kinds of numbers that can be very related but used for different reasons. Three, sometimes simple things can be more complicated than they appear at first glance.

Vocabulary:

number: a well defined abstract relationship or pattern normally associated with quantity, order and direction

Cardinality-The relationship between sets that defines what we normally call numbers.

Ordinality- an order imposed on a set

set- a group of objects or ideas that is clearly defined.

Introduction to Crash Course Mathematics

Hello everyone my name is David Rybka and welcome to "Crash Course: Mathematics." I don't technically have a right to use this title. And currently, I am making no money off of this so please don't sue me Hank Green. cue intro, For those of you who are unaware, mathematics is kind of important. It is the foundation for most of the technology, reasoning and scientific thought that we use today. So if we want to continue improving on it, it might be a good idea to learn about it. Many people find mathematics hard to understand for a variety of reasons. Large parts of how mathematics is described is in a constructed pseudo-language. It is designed to describe very specific abstract ideas that were developed over literally centuries. It makes sense that it can be a little confusing.

This course is intended to explain the language of mathematics as well as the abstract concepts behind them. I want to give you a broader understanding of mathematical concepts as well as practical methods that may help you on your homework in school if possible. The idea is to show you how these concepts are defined then show you how mathematics expresses these ideas and maybe even some of the stories and histories behind the ideas.

Mathematics is one of the oldest recorded intellectual subjects. There are cave counting sticks tools that are literally the first conceptualization of numbers. As agriculture developed, so did bureaucracy and the need to record numbers developed in what we call Mesopotamia. Specifically it was in Egypt and Babylon. This also required teaching, Egypt had the first examples of math quizzes. They also helped to develop the concept of ratio for equitable pay and geometry to define property.  Babylon had the first place value system of numbers. Greeks developed large parts of geometry and one of their number created a textbook in the subject we still use to some extent today. It is actually one of the most published and translated piece of literature other than the Christian Bible. China created the beginnings of matrices and what we call linear algebra. India developed the concept of zero in the old world (kind of a big deal as I hope to get into later). and in the Americas the Mayans had a base 20 number system that had a placeholder digit that we call zero. This lead to the number system that we use today. Which made it's way through the Islamic Middle East. Which is also why they are called Arabic numerals. The Islamic empire also gave us the beginnings of variables and what we call algebra and it is also where we get the word for "algorithm". Both algebra and algorithm are words based on the names of Arabic scholars. Then we make our way to Europe which took all of these tools and turned them into some of the most advanced mathematical subjects we have today. We have Probability and Statistics (Pascal), Calculus (Leibniz and Newton), Abstract Algebra or Group Theory (Abel and Galois), Topology (Euler), Graphing (Descartes), as well as many others.

I intend to discuss the history, the methods and the concepts of mathematics. Hopefully, in a way that is useful to those who read this. I do plan on going over basic ideas in a way that leads to some of the broader concepts behind the things you learned in elementary school. Eventually, I plan on walking through some of the more advanced ideas in math using this framework. Given the fact that this is a draft. I want people to ask me questions, ask for citation, and point out when I am wrong or confusing. I hope to have fun with this.