I normally read the good math bad math blog, where Mark Chu-Carroll debunks crackpots that try to hide their crazyness behind bad math. A while ago, he posted an article called Why Math?, where he discusses how math allows you to “without ambiguity, prove that something is true or false”.

While I agree with Mark, I think it’s not completely clear from his post the importance of math’s role in deciding what to believe.

One comment in particular inspired me to write this post.

Statements of the form “_ is impossible” strike me as deeply unscientific and dogmatic.

See, math is a formal system for thinking logically. The heart of math lies in being able to prove things from assumptions. So in that sense, if the math is done right, Mark’s remark “without ambiguity, prove that something is true or false” is spot on.

However, the assumptions need to be true or your whole argument is invalid, no matter how good the math is.

But going back to “_ is impossible”, I think there are different levels of “impossible” that we need to be concerned about.

We can classify impossible things in three categories: logically impossible, physically impossible, and technologically impossible.

## The logically impossible

To understand the first category, we need to resort to some philosophical bullshit. Back in the day when people didn’t have reddit and used to wander around in their togas thinking whether we could think at all, the great Aristotle proposed what he called his three laws of thought. These form the basis of logic and if you think about them, they make perfect sense. The first law is called the law of identity, and it just says an object is equal to itself. The second law is the law of noncontradiction and it says that nothing can both have a property and not have it at the same time. The third law is the law of excluded middles and it says that objects either have a certain property or they don’t.

I particularly like Avicena’s description of the law of noncontradiction:

Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.

This is where math helps. Math allows you to follow arguments to their logical conclusion and either prove or disprove them. It’s formalized thinking, and it’s full of little shortcuts and notation that allows you to be efficient in reaching your conclusions and in communicating those conclusions to others.

If you find something to be logically impossible, you’ve found a real impossibility. No way around it. Time travel is the best example of a logical impossibility. If you travel back in time to a place (and time) you have never been before, you would violate the law of noncontradiction (you were both there and not there). That’s why sophisticated science fiction writers like Michael Crichton always have the travelers go to a parallel universe.

## The physically imposible

Some things however, are not logically impossible, but they violate the laws of physics. This is slightly harder to detect, since our understanding of the laws of physics changes and it has been known to be completely wrong. But if you claim that something that violates the laws of physics is possible, the burden is on you to convince us that the laws of physics are wrong. Not an easy feat!

One example of this would be the claim that you can travel faster than light. Unless you’re willing to debunk Poincaré’s theory of relativity, you’re going to have trouble having anyone believe you.

## The technologically impossible

Other things don’t violate either the laws of logic (math) or the laws of physics, yet they are beyond our current reach. This is the weakest level of impossibility because it could be just a matter of time before we get there. Almost all current technology would have fallen in this category only a century ago. Just imagine trying to explain the Internet and iPhones to people of the 1800’s, they would have certainly shouted “Witchery!”.

In conclusion, statements of the form “_ is impossible” don’t need to be unscientific and dogmatic.