Understanding axioms.

LuisRodg · 2008-12-02 06:29:52

I was having a conversation with a friend of mine about absolute truth and mathematics. In this post I dont want to get into absolute truth, rather I'd like an explanation on the axioms of mathematics.

For example, are the axioms something that we take for granted or can the axioms actually be proved or do they derive or follow from something else? If its something that we take for granted then every result in mathematics would not be absolute but rather relative to the validity of its axioms?

I would really APPRECIATE if someone cleared this topic for me and gave their insights!

Thanks.

Last edited by LuisRodg (2008-12-02 06:34:01)

mathsyperson · 2008-12-02 07:40:19

You've pretty much got it in your second paragraph.

Axioms are not provable. If some fact was provable from other axioms, then mathematicians would want to call it a theorem or a lemma or something.

You're absolutely right that these axioms are basically assumptions, and so the truth of maths is not absolute. The important thing is that no theorem that we prove from these axioms will contradict any other axiom or theorem, so maths is internally consistent.

It's impossible to conclude anything without any starting information, so any mathematical system will require at least one axiom.

bossk171 · 2008-12-02 08:45:43

A really interesting book that I read that deals with this subject (and a whole bunch of others) is "Godel, Escher, Bach" (http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567).

My understanding is a axiom is like a brick, and a theorem is like a house (does that make a lemma like a wall?). Only these bricks don't come from anywhere, they just ARE. Another book that touches on this subject (but only briefly) is Flatterland (http://www.amazon.com/Flatterland-Like-Flatland-Only-More/dp/0738204420) It's a pretty quick read that explores that idea.

I'd like to expand on that question: What's the differance between:

theorem, lemma, postulate, hypothesis, axiom, rule (like Cramer's Rule), law and anything else you can think to add to this list.

Can someone smarter than me (mostly everyone here) break this down and help explore this?

mathsyperson · 2008-12-02 11:14:20

I think of it more like axioms are the floor. If you don't have a floor then you can't even begin building anything. It'd just fall down.

Lemmas being like walls is a good analogy. You get some bricks and make a wall out of them, and there you have a wall. It stands up by itself and you're not forced to do anything else with it.
It's perhaps not all that impressive though. I mean, it's a wall. It just sits there. I'd rather be looking at a puppy or something.

Then to continue from your suggestion, theorems are houses. You've got a floor to build it on, and then you start by building some walls. (Or you could wander around until you find a suitable wall that other people have built) Now that you've got a wall or two, you do a bit more work and put a ceiling on. Voila! You have a house.
And now suddenly it's very impressive and I stop looking at the puppy because over there is a magnificent house.

I should point out that houses don't necessarily need walls. When you pitch a tent, it's pretty much one step. Find a patch of floor, put up the tent. Done. (As long as it's one of those easy tents that don't need stakes and guy ropes and all the complicated stuff)

In the same way, theorems don't have to be proven from lemmas. It's fine for something to be called a theorem if it's proven from axioms alone. In fact, something that needs lemmas to be proven can still be a lemma as well. Say you've got a wall. Maybe you want to make it bigger?
You couldn't make the wall bigger if it wasn't there in the first place, but when you've finished it's still just a wall, albeit perhaps a more impressive one.

There isn't really a clear way to decide whether some statement is a theorem or a lemma though, it's largely a matter of taste on the part of whoever's labelling.
In general, theorems are useful by themselves and lemmas aren't (they're only useful because they help to prove a theorem).

Ricky · 2008-12-02 17:59:23

A part of the issue of axioms is what do we take for axioms? Axioms should be things which are intuitively obvious. For example, that a null-set exists is an axiom, or that an infinite set exists (this axiom is much more rigorous, but that's the gist). There are also axioms which allow us to say what sets are equal, or that the union of two sets is also a set.

Most axioms like this come from our human-concept of what a set actually is, and how a set works in the real world. Of course there are controversial axioms (axiom of regularity and axiom of choice). But for the most part, our axioms are things you read and then say, "Well duh...".

In standard mathematics, there are 6-10 axioms (depending on how much you put into a single axiom). Check out Zermelo-Frankel set theory if you have the chance.

Try taking out a single axiom (for example, pretend the null-set no longer exists) and see what happens. How much mathematics still holds? If things get pretty boring (as what happens when you remove the idea of "unions"), then I think it's safe to say we need that axiom.

Math Is Fun Forum

#1 2008-12-02 06:29:52

Understanding axioms.

#2 2008-12-02 07:40:19

Re: Understanding axioms.

#3 2008-12-02 08:45:43

Re: Understanding axioms.

#4 2008-12-02 11:14:20

Re: Understanding axioms.

#5 2008-12-02 17:59:23

Re: Understanding axioms.

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