I just watched this documentary about great mathematicians and their decent into madness after trying to solve profound problems, while being mocked and even hated by fellow scientists.
Being ahead of your time while being suppressed by your peers is a powerful narrative. It only happened to few people, but of course every other wacko with a messiah complex will see himself in the same light. People don’t believe me or try to shut me up? I must be ahead of my time!
But there are some lessons to be learned. Many geniuses share a characteristic with idiot savants: They are great at exploring a very small world, which happens to be only a small partition of the real one. It’s even harder to rethink the basis of your reasoning if you believe that you are already doing exactly that: Cantor was questioning the very principles of the mathematics of his time, while being deeply religious. Gödel changed the very essence of logic while assuming that he himself was somehow beyond it. These guys reached planes of knowledge I can’t even begin to understand, yet what they wanted to believe was what held them back in the end. The hostility of their peers – had none of their greatness but shared their flaws – wasn’t really their limiting factor. From the documentary, the only person whose life was made unlivable by the intolerance of the people in that time was Turing – and that had nothing to do with his work. Now while the things they discovered have long been accepted, we still haven’t learned from their errors. I guess it’s because only to geniuses these things are really unbearable. The rest of us, we just acknowledge them, and go on with our lives (having resolved nothing).
I think I can explain the things that worried people like Cantor and Gödel, even not fully understanding their actual work. For them, perfection existed somewhere. They had a preconceived, maybe native idea that everything could be understood, and that a reality where that wasn’t possible is one that is too cruel to be true. Don’t underestimate the emotional attachment to a human-understandable, in some way perfect world! Finding out otherwise is like discovering there is no god (for a deeply religious person), that your mother doesn’t like you, and that good things happen to bad people without anything to offset it. But that’s the problem: even the greatest minds let emotions cloud their judgment. That’s because emotions led them to their discoveries in the first place! No emotions – nothing that could motivate you to do great things.
So in order to discover the true face of reality – or to do so withing the possible restrictions set to us by the nature of our minds – we must first prepare ourselves emotionally. If you have any preconceived ideas about what you will find, you are limiting yourself to a solution space that might only contain local optima. Whereas if you accept the idea of an uncaring, random world where it’s not even certain that you could ever understand it – you have the best chances of finding the best possible solution. And that’s our optimum: the best thing we can achieve, not the best thing there is. Mathematicians like to make a clear distinction between the two and shoot for perfection. And then they miss. It’s probably inherent to their thinking. The rest of us can only hope that on their way (to crashing and burning), these geniuses develop a number of tools that are actually useful. There’s a lot of examples in computer science: The mathematician says “You can’t solve that problem perfectly and efficiently. It’s driving me crazy!” whereas the computer scientist says “I’m not sure if there’s an efficient algorithm for solving all problems, but here’s a subclass of problems I can solve well enough!”.
Like the documentary noted, mathematicians trying to solve a problem can behave like a computer trying to solve the halting problem. Programmers have long come up with a solution: timeouts. That would look like pure idiocy to a mathematician (you don’t really solve the problem, you just get the class of problems that are solvable before the timeout), but in the real world these things play an important role. After all, a computer will shut down at some point, just like a mathematician will cease to exist. Also, there is a very short amount of time I’m willing to wait for a program to respond. And since there’s only a finite number of algorithms that halts before a given point in time, maybe we should spend more time looking at those.
With this post already losing structure, let me repeat the points I was trying to make:
- Reality doesn’t care what you want it to be. Be it god, karma, free will, the brain being more than a machine or the human capacity to understand everything: if the only thing you know for sure is that the world look very bleak without that concept, it’s probably a figment of your fragile little mind.
- If you find yourself going crazy about what you can’t do, try to focus instead on what you can do.
Actually, the last point still needs some explanation. Let’s look at the Design Space of our universe. It contains every object that can be made using the matter around us. Now try to estimate its size. It is big enough to hold all of our inventions, with most of them working the way they should even if dramatically altered. It is big enough to hold all of us – that is all humans that ever existed or will exist. It is big enough to hold every pattern of atoms from the beginning of time up until now (and beyond). And that’s still only a small subset, if you remind yourself that a) we haven’t invented that much yet and b) most objects have been made under most severe constraints. Even the most simple organism is able to replicate – that means it contains all the information to make something similar to itself in a specific environment (that still contains a lot of variables). You can’t say that of any human invention (yet). But what is the most amazing thing: each object must come from a line of similar objects that changed only in very small increments through evolution. This seems to imply that for larger increments (as the ones made possible by inventive humans, creating things like the automobile which hardly resembles anything found in nature), the Design Space must be vast indeed. So let’s really explore it.