Toward Order

Further to this morning's post about cultural expectations for geniuses, I offer the suggestion that true revolutionaries aren't creating innovations, but discerning them in the patterns of the reality into which they've entered. Physicist Stephen Barr notes the corollary in science:

As we turn to the fundamental principles of physics, we discover that order does not really emerge from chaos, as we might naively assume; it always emerges from greater and more impressive order already present at a deeper level. It turns out that things are not more coarse or crude or unformed as one goes down into the foundations of the physical world but more subtle, sophisticated, and intricate the deeper one goes.

Barr uses the example of marbles in a box: When the box is tilted to one side, the marbles take a hexagonal pattern implied by their inherent shape. The order that we see in the packed marbles was, in a way of looking at it, part of the genius in the invention of the sphere. Such are the building blocks of all of reality.

Two responses are common from atheists or mere secularists to the species of notions of which Barr's is a member, that reality is, in fact, a divine thought: Either we happen to inhabit the one universe (of some unknowable number) in which these rules apply, thus de-necessitating God, or we happen to be privileging concepts of order and beauty that we prefer, given the universe that we inhabit. The first rejoinder doesn't actually address the argument; it merely pushes it to another level. After all, even if it took some number of universal false-starts to create our universe, the possibility of our universe must have existed within the initial concept of the multiverse.

To answer the second objection, I'll return to Barr:

Some might suspect that this beauty is in the eye of the beholder, or that scientists think their own theories beautiful simply out of vanity. But there is a remarkable fact that suggests otherwise. Again and again throughout history, what started as pure mathematics--ideas developed solely for the sake of their intrinsic interest and elegance--turned out later to be needed to express fundamental laws of physics.

For example, complex numbers were invented and the theory of them deeply investigated by the early nineteenth century, a mathematical development that seemed to have no relevance to physical reality. Only in the 1920s was it discovered that complex numbers were needed to write the equations of quantum mechanics. Or, in another instance, when the mathematician William Rowan Hamilton invented quaternions in the mid-nineteenth century, they were regarded as an ingenious but totally useless construct. Hamilton himself held this view. When asked by an aristocratic lady whether quaternions were useful for anything, Hamilton joked, "Aye, madam, quaternions are very useful--for solving problems involving quaternions." And yet, many decades later, quaternions were put to use to describe properties of subatomic particles such as the spin of electrons as well as the relation between neutrons and protons. Or again, Riemannian geometry was developed long before it was found to be needed for Einstein's theory of gravity. And a branch of mathematics called the theory of Lie groups was developed before it was found to describe the gauge symmetries of the fundamental forces.

This is where this afternoon's topic ties in with this morning's: Pure mathematics are logic crystallized, and sometimes that logic leads to peculiar and seemingly irrelevant rooms, but those who discover those rooms needn't be nonconformist radicals. What's required for effective exploration of reality, in any field, is not a bumbling and callous rebellion, but a respect for the universe and human society as we find them, and what's required for a lasting and profound change in the physical and social order is not wholesale rejection of standards, but long-seeing comprehension of the paths that they naturally take.