I really cannot get yesterday’s audacious idea out of my mind, that there is no such thing as pure mathematics. Why is it so fascinating? I guess one reason is because I have always been fascinated by the nature of intelligence. What it is, where it comes from, how it relates to being human.
I remember when I was about 15 years old, I attended my first talk about AI. It was by a man called Vidyasagar but I wish I could remember more about him. He asked the question, if a computer program comes up with a new theorem in mathematics and proves it, where is the intelligence? In the computer? In the program? In the programmer? In the person who proves that the program works? And he went on to say that his own opinion was the last. It struck me as dreadfully strange. I was to bashful at that time to ask him why, and hoped someone else would, but no one did. So I never knew…
Now, however, I have a slightly better understanding about the nature of mathematics, so I can guess why. Let’s use Egan’s argument as a starting point. In Luminous, he asks (and I have asked this before), could there be a physical system that constitutes a proof of, say, Fermat’s last theorem? A certain configuration of billions of quarks, perhaps, that by its very existence proves it? Egan’s answer is, yes; if nothing else, a certain configuration of neurons in Andrew Wiles’s head is the physical proof of Fermat. This is still a bit bizarre for me to wrap my head around, but if true, then intelligence being defined as the place where the proof dwells is somewhat more acceptable.
The corollary, and this is another of Egan’s favorite themes, is that intelligence is a configuration. It is a certain state of matter and energy, a pattern, that doesn’t have to be spatially or even temporally contiguous. (This is the theme he explores in such detail in Dust.) This, too, mirrors my own belief. Do you remember that scene in Hofstadter’s GEB, where a character called Aunt Hillary is introduced? She is an intelligence that emerges from an ant hill. The ants perform a function in the anthill that mimics neurons in our brain, each ant having no intelligence at all but by their patterns being able to represent thoughts as deeply as any human.
The problem with recognizing this as intelligence is the limitation that we cannot communicate with it. Hofstadter introduces some whimsical way in which this anthill communicates with the world through pheromone trails, but I don’t remember the details.
I do remember thinking, though, that this mirrors a long held belief of mine, that trees are intelligent, albeit in a timescale and space scale that are so different from us that we cannot communicate with them.
At least, not yet. But as we slowly come to terms with artificial intelligence, I have faith that we will establish ways of communication with the millions of non human intelligences that lurk in nature, too.