# What are Common Notions?

## In other words "axioms"

﻿In this section we actually get to see true axioms (postulates) in the way we define the word today. An axiom is a statement that is true without proof; the truth in an axiom can come in different ways. ﻿For example, you can have logical axioms like "A statement and it's opposite cannot both be true simultaneously." A door is either open or closed (we will consider "closed" to be completely latched and anything other than that will be "open" even if you can't actually fit through the door), so we can logically say that a door cannot be open and closed at the same time. We don't have to prove this, we know it and understand it from how we defined our terms and by using basic logic. ﻿The second type of axiom is non-logical and this is where things get interesting for math. For this section I will keep it simple but if you like to read more of my ramblings, just see the next section below. For non-logical axioms these are statements that are true because we say that they are, they don't follow from any source of logic or common sense but they must be true. In fact, axiom comes from a Greek word that means "to deem worthy." These are the foundation stones from which we build mathematics. For example, lets look at some of the Peano Axioms for natural numbers: 1. 0 is a natural number 2. For every natural number x, x=x. Equality is reflexive 3. For all natural numbers x and y, if x=y, then y=x. Equality is symmetric 4. For all natural numbers x, y, and z, if x=y and y=z, then x=z. Equality is transitive ﻿Those words are probably familiar from middle school pre-algebra and you probably though this stuff was common sense, of course a number must equal itself, right? But why? Why is 0 a natural number? It doesn't have to be, in fact when these were written it didn't say 0, it said 1 instead and it worked just fine, we changed it later just for consistency. These statements create a basic framework for math in the natural numbers, they aren't true because the universe says they are, they are true because we want them to be and they make this thing we created called arithmetic systematic and we have found that arithmetic is very useful for our lives. Math that is useful to solve problems we tend to keep around, while not so useful math tends to fade away (look up Lunar Math). We could imagine a system were the above statements are completely different, no problem, the question would then be "Are these new statements useful?" Sometimes it can take centuries for that question to be answered and one new development makes what seemed like an insane idea become the cornerstone for a new type of mathematics. ﻿For our purposes, we will be using non-logical axioms in this section.