Welcome to the philosophy category, where I talk about the meaning of life and other words. This particular post is based on a intro philosophy class I am currently taking, and on my own prejudices.
In philosophy, all knowledge is divided into two categories. Those categories are "a priori" and "a posteriori". Over history, the line between these two categories has been drawn in many different ways, but I'm going to talk exclusively about the definition of our own era (what you might call the post-modern era). Funny, when we covered all the various definitions of "a priori" throughout history, this was the one that came most intuitively to me. Perhaps I am just a product of my own time.
A priori truths are propositions that absolutely must be true, just based on the formal definitions of the words. A posteriori truths are those that require justification by actually observing the universe. It seems to me that the definition of a priori is pretty close to "things that require only deduction, no induction."
Here's an example of an a priori truth:
All bachelors are unmarried.
It absolutely must be true, just by definition of "bachelor." More generally, you can make up any word, define it, and prove certain things just by definition. For example:
Let an "ultra-blog" be any blog with more than one reader. If my blog has more than one reader, then it is an ultra-blog.
Of course, I chose a rather non-intuitive definition there, since you would think that "ultra-blog" would refer to something much more amazing than mine, but I'm completely within the rules here. Therefore, if I look into the world and find I have readers besides myself, I will have proven a posteriori (since it required looking into the world) that my blog is an ultra-blog.
Math is made up of a priori truths. For example, "2+2=4" absolutely must be true, just by assuming the Peano axioms (explained so well by MarkCC), which basically define natural numbers and addition. Certain philosophers try to reject the idea that math absolutely must be true, claiming that it's only true by nature of our universe or something, but I think that's wrong.
You might say that math requires certain axioms that we assume to be true, when they might not be. I might fix this problem by changing the statement to "If the Peano axioms are true, then 2+2=4." But I won't, since that's as silly as changing the bachelor statement to "If the definition of 'bachelor' is an unmarried man, then all bachelors are unmarried." Axioms are just like definitions, only they define mathematical systems (such as arithmetic) rather than specific words. Assuming that axioms are true is really no different than assuming that words mean what we define them to mean. Unless we're using unusual words or unusual mathematical systems, I think it's better to leave the definitions and axioms implicit.
So in conclusion, I think there's one thing that we can be absolutely sure of: words mean what we define them to mean--by definition.