As a companion piece to yesterday’s post have a look at this essay in the religious periodical First Things, written by Amanda Shaw. The purpose is to draw a parallel between imaginary numbers and belief in God. You see, for centuries mathematicians scoffed at the idea of imaginary numbers, but a few brave folks were able to look beyond the stiflng orthodoxy of their times and now imaginary numbers are commonly accepted.
See where this is going?
Shaw presents a passable history of imaginary numbers. But if her intention is to develop a parallel between belief in imaginary numbers and belief in God, I’m afraid she has missed the most important parts of the story.
Here’s Shaw’s opening salvo:
Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary: You can’t read much about religion today without encountering these adjectives, each intended to leave religious belief with a tired, messy, belittled sort of look.
You see it most with those who claim to be speaking in the name of scientific reason. The history of science itself, however, shows that the best scientists do not limit reality to the tangible or even the comprehensible. Unafraid of wild speculation, these are the minds making breakthroughs, thinking what was once un-thought, discovering what was deemed impossible. Scientific positivists may scoff at the irrational and the imaginary. But science does not.
We’re off to a bad start, I’m afraid. Imaginary numbers are those whose squares are negative. Within the reals such numbers are a logical impossibility. So how did it come to pass that such numbers became a mainstay of mathematical practice?
As Shaw tells it you picture a stifling orthodoxy dogmatically asserting the impossiblity of such things, thwarted by the clear-thinking and open-mindedness of a handful of geniuses. Alas, this is a far too melodramatic reading of events. It is not as if a handful of mathematicians were saying, “Please, please let us take square roots of negative numbers!” while the short-sighted, unimaginative establishment came down on them and forbade it.
What really happened is that mathematicians saw over and over again, especially in finding solutions to certain polynomial equations, that square roots of negative numbers kept coming up. Everyone understood that within the real numbers it is simple gibberish to write the square root of a negative number, just as it is gibberish to divide by zero. Nonetheless, a few people noticed that if such square roots were manipulated as formal symbols according to the standard rules of algebra, meaningful statements could be obtained from them.
Shaw does a decent job in recounting this history, though I think she embellishes some things here and there. Let us skip ahead to the end of her essay:
Scientific positivists, pencil and paper in hand, peer through shatterproof, UV-protected glasses at a world of animals, vegetables, and minerals. But genuine scientists–true seekers of knowledge–are not afraid to let the sunlight dazzle them, not afraid to seek and imagine what our myopic reason calls absurd.
Impossible, irrational, delusionary, absurd, untrustworthy, fictitious, imaginary: It is always easier to approach–or rather, ignore–mysteries of math by dismissing them as false or unintelligible. And how much more for mysteries of faith. So is God like an imaginary number, waiting to be discovered and accepted in a renaissance of faith? The simile is ridiculous, on its face. But, in a curious way, the ramblings of scientific history remind those who strive for reason just how vast reality is. The realization is at once unsettling and exhilarating: Truth is far richer than our minds–always confined by the here and now–can prove or even imagine.
I’m not sure if Shaw fully appreciates just why the simile is ridiculous. Complex numbers (of which the pure imaginary numbers are a subset) earned their acceptance first by proving their usefulness over and over again, and second by being placed on a firm logical footing by various mathematicians. Nowadays complex numbers are not one wit more mysterious than real numbers or rational numbers or even just ye olde counting numbers.
It was logic, reason and hard work that showed the need for complex numbers, and more logic, more reason and more hard work that transformed them from gibberish into a useful tool.
Compare that to God. Back in the days when everything in nature was mysterious and unpredictable, it made sense to invoke unfathomable gods to explain it all. But the march of scientific progress, far from showing the need for invoking God as an explanation, has actually gone in the exact opposite direction.
People begrudgingly accepted the counter intuitive existence of complex numbers because they were forced to consider them in the course of other investigations. On the other hand, science has consistently shown the obvious explanation, “God did it!” to be unneccessary and unhelpful, but people insist on clinging to it anyway. I’d call that a point of disanalogy.
Furthermore, complex numbers are rigorously defined objects that are manipulated according to clearly defined rules. God, by contrast, is routinely described (here included) as being beyond reason, something that can not be quantified or measured or tamed by mere logic. I’d say that is another point of disanalogy.
Shaw is just playing word games here. Mysteries of math and science are things that get resolved by logic and hard work. Mysteries of faith, whatever that means, are not. Mathematicians and scientists frequently hypothesize the existence of things beyond our senses, but they do so only when the evidence clearly points in such a direction, and subsequently work very hard to understand these unseen, unfelt entities. The existence of such entities may be controversial for a while, but eventually the evidence and data reaches a level where every properly trained person can come to a reasonable conclusion on the matter.
Theologians, by contrast, take God and his major attributes as given, then tediously try to fit the facts of the world into these preconceived notions. Far from revelling in mystery, the world’s major organized religions claim to have the answers to the meaning of life and our fate in the world to come. People who doubt the existence of God are never given evidence to persuade them. Instead they are treated with scorn and derision, and are often told their doubts will earn them an eternity in Hell.
I’m afraid I see no useful parallel to be drawn between these activities.