Wednesday, September 21, 2011

Math doesn't lie... Right?

If a bucket has 2 apples in it, and you add 2 more apples, how many apples are now in the bucket?

4 right?

2 + 2 = 4

"Math doesn't lie." Right?

That's true, "Math doesn't lie." But math CAN be wrong.

What if I told you the correct answer is 6?

Would you cry foul and declare your original answer to be true? Would you show me mathematical theorems and proofs that demonstrate that 2 + 2 DOES in fact equal 4?

As I said, "Math doesn't lie.", but math CAN be wrong when it tries to represent the real world. Or, let me rephrase - we are wrong when we assume that the world described by the math, identically represents the world in which we live.

In our scenario, that math didn't lie. In fact, the math wasn't wrong. But we were wrong when we assumed that the math accurately represented the world. What the math failed to take into consideration was that John also added 2 apples to your bucket. Thus, there are now 6 apples in your bucket.

"That's not fair! You never mentioned anything about anyone else." 

That's true, but when we create math to model the world, we aren't TOLD all the characters and variables and the exact scale and nature of their effects. We have to derive them ourselves. Sometimes, in simple systems, we can deduce all of them and the math does accurately reflect that system. However, the more complex the system, the more variables, the more interactions, the more complex the math, the more difficult it is to discover all affected and affecting variables, and the easier it is to miss (or misunderstand) something. Thus, when we create mathematical models to represent extremely complex systems, the potential for overlooking variables (or even simply under or over estimating their effects) is not only possible, it's likely.

In the world described by the math, John doesn't exist. Nobody other than you exists. Therefore, if you only added 2 more apples, there will only be 4 apples in the bucket now. It's simple, it's straightforward, but it'd be wrong. Overlooking John as a variable means that, while the math didn't lie; while the math wasn't wrong, it did not reflect reality. And if we had based our actions on the result of that math, we would have potentially made the wrong decision.

Let's say that we adjust our math, represent John, and based on this math, decide we have enough apples to bake an apple pie. However, what happens to our plans if, because of our mathematical oversight of not considering Sally, we only wound up with only 1 apple in our bucket because our math didn't represent that she removed 5 apples for herself? Based on our math, we would have concluded that we had enough apples, and we would have began preparing to bake. However, at some point, we would have come up very short of apples.

Similarly, when we try to model complex systems like climate or the economy, we can easily draw the wrong conclusions and plan the wrong actions based on, not faulty, but incomplete, math.

Even looking at previous data can be misleading if not all variables are considered. If we look at data that shows that home sales went up at the same time that tax rates went up, we could draw the conclusion that higher taxes cause increased home sales. Based on this conclusion, we could plan to raise them even more in hopes of further increasing home sales. However, by overlooking that a new factory opened up a few miles down the road, our conclusion would be wrong, and our actions would be mistaken. In the end, our wrong conclusion and the following actions might have led to not only the reduction in home sales, but also possibly the closing of the factory.

This type of oversight can happen easily. Looking at some specific economic data, one can easily see how people would conclude that getting into WW2 got us out of the Great Depression. However, the data doesn’t tell the whole picture. Similarly, many economists and politicians will bring up a chart or point out some data that indicates that their prescribed action worked in the past and should be implemented now. But without taking a full, contextual view of that event and that time, it’s difficult to understand all the variables that may have affected things.

“So are you saying we can’t trust math? Well we might as well throw it all out the window?”

My point isn't to say we should ignore math. My point is that we need to be cautious about just blindly trusting what math "tells" us. We need to understand that in complex systems, even a small misunderstanding of a single variable, let alone possibly omitting one entirely, can change a negative feedback system into a positive feedback system. We need to be careful about basing actions on mathematical models without trying to look at the entire context of the situation. Otherwise, we may very likely find out we don't have enough apples.

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