I've always wondered what would happen aliens suddenly came to Earth, and then, as an ultimatum, declared that humanity would be spared if and only if one human was taken and put in some kind of stasis field for eternity -- literally forever. What that one person would see would be an endless white expanse, so he/she would have nothing to but to sit and think. Forever. And I always imagined what would happen if that person was me.
That's kind of a disturbing thought, to be trapped like that, with nothing to do. It's sort of a childish flight of fancy (but quite intriguing to think about, I have to admit), but these days I've been sort of using this as the foundation for a thought experiment.
Recently, I've come across several articles aluding to the possibility that mathematics, the traditional abstract, ideal, removed-from-the-universe branch of science, was actually just another observational science, like astronomy or biology. Now that's preposterous, you say. Mathematics is one of those lofty pillars of human thought that transcend observation and experimentation -- a truth of mathematics is immortal and can be derived independently of any physical observation, right?
This brings up the question of "What is derivation?" Plato believed that mathematical objects already "exist" (in that realm of universals), and we are merely observing "instances" of these ideal objects. So are we truly discovering mathematical truths when we prove something? Or are we simply observing what is already there in the universe?
Mathematicians like to believe that there is some special, unique faculty of humans that allows for that jump of insight, that flash of intuition that leads them to epiphanies about the mathematical universe. Most junior-high students taking plane geometry will tell you it's damn hard to figure out proofs. Indeed, to come up with a logical pathway to some truth in question seems like the act of creativity, which we believe to be a quality unique to humans. It's that feeling that humans are being creative that leads to the conclusion that, indeed, humans are creating mathematics. Proving a conjecture is as much a process of creation as painting.
My thought experiment: would this man, stuck in eternity with nothing but his logical faculties intact, be able to derive -- no, create -- all of mathematics? And from there would he be able to formulate the laws of physics exactly as they are in our universe? He would become a God, in a sense, creating the universe in his mind.
The idealist in me says "yes." I (wish to) believe this because I like the idea of the universe's existence being contingent only on the rules of physics, those being contigent on the rules of mathematics, and those being contingent on the irrevocable laws of logic. In other words, the universe is ideally perfect.
But it is hard to believe that this is so. Much of our mathematics comes from physical observations. We hear of Greek mathematicians (such as Pythagoras) verifying the now-irrefutable laws of triangles and geometry by scratching very precise diagrams in sand. Newton invented a branch of mathematics dealing with infinitesimal sums and limits to explain the cosmos. It seems that for most of history, mathematics has been tailored to the beck and call of physics.
(Slight digression: I wrote Newton "invented" calculus. The subtle question here is: did he really? Or did he merely "observe" the process of calculus taking place in the universe, and put it together in a conceptual framework that humans can understand?)
It is only a recent development of mathematics (past 2-3 centuries), it seems, to deal with the incredibly abstract that have no direct connection to physical reality, such as number theory. Number theory is a sort of a "meta-mathematics" - a formalized logic system to verify the validity of mathematics itself. Can you really call it mathematics? Perhaps we should really divide mathematics into two camps: mathematics used to describe and model physical reality, and the mathematics used to describe the first. It seems that one is yet another framework attempting to model the previous one.
So here we have sort of a hierarchy (top-down):
- Physical reality (as is), which leads to:
- Physics, man's capacity to model physical reality using:
- (Physical) Mathematics, which is described by:
- (Logical) Mathematics, which is based on the pillars of:
- Logic, which is __________
My question is: in the universe, is the hierarchy reversed? In other words, does simply having the foundations of logic give growth to everything else?
More to come in Imagining the Universe, Part 2