No Can Do

Book #21 (June 17, 2010): Beyond Reason: Eight Great Problems that Reveal the Limits of Science by A.K. Dewdney

A.K. Dewdney used to write a column for Scientific American called Computer Recreations. I remember that I enjoyed reading it back in the 80s, when I was obsessed with computer programming and liked to learn about the more esoteric aspects of it. He also wrote an interesting little book called The Turing Omnibus (the title is a pun that I’ll refrain from explaining), which had some fascinating things to say about algorithms and problem solving. I’d forgotten all about Dewdney, though, until I stumbled on his name somewhere the other day and wondered if he’d written any more books. It turned out that he had and the title above was the most recent. In an attempt to reawaken my interest in science and computing, I decided to read it.

What it’s about is limitations, things that humans and science can’t do or can’t know. These aren’t technological limitations but real limitations imposed by the laws of the universe. No matter how good our technology gets or how bright our computers become, these limits will always be there (though in some cases there may turn out to be ways to work around them). The best known of these is probably the speed of light barrier, which Einstein established with his Special Theory of Relativity. The speed of light barrier isn’t some sort of cosmic signpost that says we’ll get arrested if we exceed 186,000 miles per second. It’s a real physical barrier caused by (among other things) the fact that we gain mass as we gain speed. If we could move at the speed of light our mass would become infinite and so would the amount of energy required to accelerate us. Thus, 186,000 miles per second is as fast as we can ever go.

As the title suggests, there are eight such limitations discussed here and I’ll list them with merciful brevity:

1. You can’t build a perpetual motion machine — You can, in theory (and in the absence of friction), build a device that runs forever without additional energy input beyond an initial push, but you can’t get it to do useful work, like generating electricity or grinding grain. The energy transferred into the work would eventually bring the device to a stop unless more energy were put into it.
2. You can’t move faster than the speed of light — See above.
3. You can’t simultaneously detect the location and velocity of a subatomic particle — This is otherwise known as Heisenberg’s Uncertainty Principle and has some very spooky ramifications.
4. You can’t come up with precise predictions of the future behavior of systems that exhibit so-called “chaotic” behavior, like the weather — At best you can come up with an educated guess, because even the tiniest imprecisions in your data can destroy the prediction completely.
5. You can’t square a circle — This has to do with creating one shape that has exactly the same area as another shape using only a t-square and a compass. You can do it with most shapes, but you can’t use this method to create a square with exactly the same area as a circle. Why you’d want to, I’m not entirely sure, but apparently it could be useful under some circumstances.
6. You can’t prove every possible axiom in a mathematical system — This is otherwise known as Gödel’s Incompleteness Theorem, which was a real blow to mathematicians in the 1930s, because mathematicians of that era liked to believe they could do anything.
7. Some computer problems can never be solved — There just isn’t an algorithm for everything. Sorry.
8. Some computer problems that have algorithms would take too long to solve — This is largely because the complexity of the problem grows so rapidly as you add elements to it that the fastest computer imaginable would take more than the lifetime of the universe to solve them.

Dewdney is a clear and often witty writer, but this book exists out on that scary edge of popular science writing where math starts to appear. To Dewdney’s credit the math almost never goes beyond high school algebra, but there’s a lot of it and my algebra is very, very rusty. So I very quickly started glancing at the equations and just taking Dewdney’s word for what they proved. Unfortunately, I lost a lot of the value of the book this way — he presents mathematical demonstrations of some very important stuff here — and I can’t recommend the book to anybody who isn’t willing to sit down with a piece of paper and work through the math step by step. However, even if you don’t do this, there’s a lot of interesting history here, about perpetual motion hoaxes and rivalries among mathematicians, to give the book some value even beyond the math.

Fortunately, I was already aware of most of the subjects he discusses, from the reading I did back in the days when I wrote science books for kids, so I was never completely lost, but I did develop quite a headache from staring at all those numbers.