Were the world’s most “difficult” math problems constructed to be unsolvable?
In: Mathematics
No, not intentionally. Usually.
Most of the “hardest” math problems are such that the phrasing of the problem is rather easy, but the solution remains frustratingly out of reach. On the other hand, there are innumerably many insanely difficult, yet solvable, math problems that you will never hear about. These types of problems are typically just very dense with calculations that are hard to perform. For that reason, let’s just stick with the millennium prize problems.
Almost all of the millennium prize problems came from a very intuitive observation of facts. Some are a bit less obvious, but all of them could be understood in a few days of intense study. It is for this reason that they are so tantalizing – a layman could understand the problem – and the likely solution – in a couple of days; yet the brightest mathematical minds of the last century have only been able to crack a couple of them. You can look into it the history of some of these problems on your own, and you might be surprised just how simple the problems are to understand. The genius required to identify the problem is clear, yet the solution is counterintuitively impossible to divine.
Personally, I think the Navier-Stokes equations are the best example. The equations describe how fluids work. Scientists use the equations to model real world systems all the time, and we have analytical solutions to some systems. But there is no general solution that we can find, and stranger still, we can’t prove that a general solution doesn’t exist. The icing here is that this is a mathematical model for the real world! The solution must exist. But here we are.
The solution must exists. We know it must. But we can’t prove it. The difficulty isn’t intentional. Our math just hasn’t caught up. Or something. We just don’t know.
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