Reasoning: A Math Parable

(This post is part of series of posts, beginning here. It is recommended they be read collectively, and in order.) 

Let's start this conversation on Reasoning with a parable, in mathematical form:

Let's assume we have two numbers - they could be any two numbers. We will call them 'a' and 'b'.

Now, for the purposes of this parable let's assume a=b

If we multiply both sides of this equation by a, we have axa=axb

which can also be written as a²=ab

now, let's add another a² to both sides: a²+a²=a²+ab

we can represent that more simply as 2a²=a²+ab

Now, subtract 2ab from both sides:   2a²-2ab=a²+ab-2ab

again, simplify: 2a²-2ab=a²-ab

and simplify further: 2(a²-ab)=a²-ab

Now you will notice we have a common term on both sides: a²-ab

So we can remove it, and we are left with 2=1.

And just like that, the world economy collapses, the universe is thrown into utter chaos and disappears in a puff of logic.

How can this be? What sort of devilry is this? Did I get one of those steps wrong? Add too many 2's or a's to one side of the equation?

No, it is nothing of the sort. Every step is written correctly. The problem is, while going through all those steps, through all that complexity, we forgot something important.

In the very beginning, we stated that a=b.

and so a²-ab is really the same as a²-a²

and a²-a²=0

so that last bit is 2x0=1x0

which is true, since anything multiplied by 0 is 0

And quite suddenly, we can see the whole equation was absurd. And the complexity of the equation- it's many steps, obscured the obvious.

The universe is a large and complex system, with many variables, and as such, we may be tempted to see complicated answers to every problem. It is critical however to examine such complex answers against the more simple, fundamental observations. Quite often, we add unnecessary complexity which obscures a much more obvious answer, and thus prevents us implementing a simple solution.