What are Common Notions?
- Mathew Purser
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.
If ever a mantra described math in my mind, it would be "Reality can be whatever I want." Most people are familiar with the scientific method: Identify a problem, form a hypothesis, experiment, analyze data, adjust hypothesis, repeat. science is our way of explaining our universe and science that does not explain what we see and experience is not useful and we adjust our understanding or completely abandon the idea. For centuries people believed that odors caused disease (if you ever wondered why plague doctors are depicted with those long bird masks, it was a breathing tube filled with sweet smelling herbs to "kill" the odor and prevent disease) but when we studied the problem and put those ideas to the test, it fell apart and we realized that microorganisms not only existed but had the power to kill. Science must come from reality and we write laws to say what things happen and theorems to explain why things happen. Math... Math is a different story. Science is discovered and math is created, like art. There is nothing about the universe that we studied to find math, it doesn't exist, even numbers are made up. Think of five, what do you see? The symbol 5? Maybe the word "five"? What if your language doesn't use the Latin alphabet? What if I asked you to show me five, how would you do it, with an open hand? But that's not five... yes it's five fingers but I asked for the thing that is five. I always enjoy teaching complex (nonreal) numbers to students and listening to the "why do we have to learn about something that's not real!?" and then asking the questions I just asked you about 5. Why do we accept 5 as "real" when it doesn't seem to exist while 4+2i is banned to the realm of "nonreal." Now I do want to clarify, math is real, it is just not a physical thing, it's an idea. It can be whatever we need it to be in this moment (as long as it is logical) but the beautiful thing is (and yes, math is incredibly beautiful) that when we develop math on our own, completely in our minds we can typically find a use for it in the real world but sometimes we can't and that's fine, it's still real. Science is like a tool maker looking for problems then designs a tool to solve them. If the tool doesn't work then it either throws it away or modifies it until it does work. Math is like a tool maker who sits in his workshop all day and makes random tools, knowing one day, someone will be smart enough to figure out what problems they solve.