Tuesday, April 3, 2012

Inductive reasoning, and why you should care

Inductive reasoning, according Wikipedia:

Inductive reasoning, also known as induction, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations of individual instances of members of the same class.

That probably conveys absolutely no value to you. The best way to understand what inductive reasoning is, is to apply it.

In order to do that, we need some kind of problem which we can solve using a bit of induction.

I'm going to a pick a math problem. Why, you say? Math is almost always the easiest way to understand reasoning, since math's primary device of "progress", so to speak, is reasoning.

So, here's the problem:

The plane is divided into regions by drawing a finite number of straight lines. Show that it is possible to color each of these regions red or green in such a way that no two adjacent regions have the same color.

If at first this doesn't strike you as a math problem, you probably haven't met this chap called graph theory, or maybe haven't been introduced to bipartite graphs.

So, how do we go about solving this?

Before reading on, try out some cases; one line, two lines, ten lines, and so on. Try and figure out some patterns.

Now, I'll show you the solution.

First we consider one line. So, we can color the regions on opposite sides of the line opposite colors (i.e. red and green), and, we're done.

So, what happens when we add one line? Think about this for a moment. Can we try the same trick?

Yes! What we can do is take one side of the new line we just put in, and swap all the colors on that side (so, red goes to green, green goes to red), and, we get a complete solution! So, every time we add a line, we can simply follow this algorithm and get the correct coloring. Of course, this isn't a fully rigorous proof, but, it shows what we're after.

Inductive reasoning is taking a method you used to solve a problem for a set constraint (we knew how to do it for one line), and then extending that method to solve the problem without the constraint.

If write code, you probably do this all the time (and just didn't know what this was called). If you've got a function that's misbehaving, you pass in a couple of values that you think are forming an edge case, and you've noticed that its going wrong for *one* of these values, you change the function definition so that the errors don't happen.

But, you should use this kind of reasoning much more often. In nearly all problems where you are stuck on something going wrong on a constraint, try inductive reasoning. Of course, none of this actually happens consciously, but, if you try doing it consciously a couple of times, it "just happens" eventually.



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