How Math Doesn't Get Done

Say I have a theorem I want to prove because its given in an exercise set. In this case show that,

│x₁ + x₂+...+x_n │ ≤ │x₁ │+ │x₂ │+...+ │x_n │

and my next thought is that I need a “good” definition. How do I decide which definition to use? I find a bunch and I think they ought to be good definitions because I looked them up in a book written by someone with a degree in math.

Definition 1: │a│ = { a, -a } ⁺
Definition 2: │a│ = max{ a, -a }
Definition 3: │a│= a sgn a
Definition 4: │a│ = sqr root of a²
Definition 5: The absolute value of a real number a is defined to be a if a is positive or zero, and to be -a if a is negative.

Day Two:

But on the other hand, what if I don't have a book and I'm online or I get a bunch of trash from wiki. How am I supposed to know a good definition when I see one?

And say I get in the middle of a logic book to figure this out and I start reading the chapter in the middle of the book on “Theory of Definitions” and it makes a lots of references to Aristotelian logic, then out comes the Aristotle and see that I have at least 350 pages to read if I want to make any sense of it. I get back online and look up what I can about the original theorem. Why is it given in the exercise set? What makes it important? It doesn't look like the rest of the problems in the exercise set so it must be special.

Day Three:

So I go back to wiki and there is a discussion about the “properties” of absolute value and one of its properties is “subadditivity” which, lo and behold, looks exactly like the problem that I have in my problem set. So is there something important about subadditivity? By the dogs, it seems there is something to this subadditivity thing. All sorts of things can have this subaddtivity property, but most importantly, it seems a distance function must have this property....(What if it doesn't?) What are the other properties that a distance function must have? Is a distance function the same thing as a metric? Is a metric the same thing as a metric space? No. it seems not. A metric space includes a distance function. What sorts of distance functions are there other than absolute value?

Day Four:
So while I'm thinking about what makes a good defintion good, it occurs to me that perhaps illustrations are not definitions. Are illustrations actual objects? In geometry an illustration of a circle is not the actual circle we are discussing, it's an idealization of some circle, but wait a second, there aren't any actual Platonic circles floating around anywhere, so what can an illustration of a circle really represent?

Day Five:

This inequality is like the triangle inequality but for a triangle with an infinite number of sides...er sort of...like a circle? No. Must stop thinking in pictures. Maybe Euclid could be done without any pictures at all, maybe circles aren't really circles but sets of points, lots of symbols, what would Euclid look like if it was written in set notation? (Answer handed down from above: Don't worry, Euclid defined a circle, you're safe, they aren't just illustrations).


Day Six:
And when I look at a kids textbook and it says that absolute value is "the" distance, is it really the distance or is it a way of measuring that distance? I mean, the actual number yielded is the distance, but the thing that got you that number, the absolute value thingy itself does not seem to be "the" distance. See what I mean? So maybe, since there are other distance measuring thingies like absolute value,which I have no idea what they are but wiki or some other math site said it so it must be true, maybe I should say, "Absolute value is one way of measuring distance." I'm having issues with the idea of measuring distances on non-physical objects.

Six days later, I'm finally finished with my little proof, but if I'm going so slow because I can't really focus on the actual problem then it's going to be forever before I learn anything.

But before I go on to problem #26 let me just check this one little thing out about this Fekete's Lemma. Wonder what that's good for? And if I could find out would I even understand? It's not in any books around the house but it looks interesting.