Our ancestors were as smart as we are, and when it got dark at four o'clock in the afternoon, the other thing to do was play with the insides of their own heads. Understanding some puzzles from antiquity as the thinking of past mappers is useful not only because it is interesting, but because it shows us what the unaided human intellect is capable of. This is something we need to appreciate if we are to regain control of our work from the processes we have handed our lives and careers to.
Infinity was a hot topic, and our ancestors had broken this notion down into three different kinds. Conceptual infinity is easy - you just say
forever' and you've got it, for what it's worth. Next, there is a potential infinity. You can give someone an instruction like,Keep counting forever'. In theory, you could end up with an infinite collection of numbers that way, but could it ever really happen? Could you ever actually get an infinite number of things right in front of you, to do amazing conjuring tricks with? They realized that if an infinite collection of anything from cabbages to kings really existed, it would take up infinite space, so if there was an infinite collection of anything with any size to it, anywhere in the universe, we wouldn't be here. There would be nothing but cabbages - everywhere. We are here, so there is no infinite collection of anything with any size to it, anywhere. But there is still the possibility of an infinite collection of something infinitely small. If something can be infinitely small, then God (who is handy to have around for thought experiments because he can accomplish anything that can be done in this universe) should be able to get an infinite number of angels to dance on the head of a pin.
Our ancestors felt that this idea was ridiculous and that therefore there is no actual infinity in this universe. Today, we have two great theories of physics. One works at large scales and uses smooth curves to describe the universe. The other works at small scales and uses steps. We haven't got the two theories to mesh yet, so we don't know if the deeper theory behind them both uses steps to build curves, like a newspaper picture, or if it uses curves to build steps, like a stair carpet. It might be something we've not imagined yet of course, but if it's one or the other, our ancestors would guess the steps, because of the angels on the head of a pin.
What about the dragons? They roar and belch flame. Their noise travels faster than the wind. They collect precious jewels below the ground. They live in South America, China, Wales. They eat people. They are worms, and an ancient symbol for the world is the great world worm. They are a conceptual bucket in which our ancestors gathered together what we now call tectonic phenomena. They had no idea that the world is covered by solid plates wandering around on a liquid core, but they had eventually gathered all the effects together through mapping applied to direct observation. The dragon took the place of the real thing in their mental maps until by wandering around they discovered the real phenomena that produced the effects they tagged `dragon'.
And alchemy? The futile search for a procedure for turning base metals into gold and getting rich quick? An alchemical or Hermetic journey consists of a series of operations (which may or may not have physical manifestation such as a diagram or experiment or maybe just a thought experiment), performed by the operator. The journey ends at the same place that it begins, and during the journey, the operator's perception of the world is changed. The operator's consciousness has been deepened and enhanced, and it is he, not the stuff on his desk, that is transformed. The return to the beginning is necessary because it is only then that he sees that that which was obscure is now clear. Alchemy is mapping.
In the great cathedrals of Europe, there are many arches holding up the roofs. In these days we'd probably get a symmetric multiprocessor to grind out a finite element analysis, but the builders didn't have the hardware or the algorithms. They didn't have the nice equations we have in mechanics or even Newton's own Latin prose. Most of them were illiterate. But if you compute the optimal arch strength/mass curve for the spans, you usually find they were bang on. They did this with the only tools to hand - their own experience, and the ability we have to get a feel for anything with the neural net between our ears.
Make sure you have a realistic evaluation of your own capabilities. The usually necessary correction is up! Getting good at anything takes practice, but given that you'll be doing the work anyway, it's nice to know how good you can get.
Creative hacking and responsible engineering are orthogonal, not contradictory. We can have the pleasure of stretching our faculties to their limits, and yet fulfil our obligations to our colleagues.
Copyright (c) Alan G Carter and Colston Sanger 1997