Is God Like an Imaginary Number?

Posted by jrosenhouse on September 4, 2007
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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.

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Comments

  1. #1 Stuart Coleman
    September 4, 2007

    I’m afraid I see no useful parallel to be drawn between these activities.

    And I can’t fathom how anyone does.

  2. #2 RBH
    September 4, 2007

    “… and they also laughed at Bozo the Clown.”

  3. #3 Tyler DiPietro
    September 4, 2007

    It’s amazing how often IDers attempt to skip the effort of demonstrating the usefulness of their ideas by rhapsodizing about how their struggle is analogous to some once disfavored concept, not matter how little the scenarios they invoke haven in common with their pseudoscience.

    And not to put too fine a point on it, but it is worth mentioning that all of the 15th. century mathematicians that initially scoffed at imaginary numbers were Christian theists, not stodgy atheists.

  4. #4 Mr. Language Person
    September 4, 2007

    . . . not one wit more mysterious . . .

    whit

  5. #5 Coin
    September 4, 2007

    Well, I mean, I don’t know. Can God be used to solve cubic equations?

  6. #6 noncarborundum
    September 4, 2007

    Obviously you missed the History of Mathematics course that Shaw took. It’s quite well attested that the oldest mathematical manuscripts from the world over are full of a variety of imaginary numbers interacting in fantastic but edifying fashion, following which mathematicians in the Ancient Near East developed the idea of a single, super-complex number than which no number more complex can be conceived. This number began to be considered extraneous to mathematics only sometime in the last couple of centuries, when mathematicians discovered that their calculations did not require its existence. Nowadays only fellows of the the Institute for Complexity Research (ICR) and affiliated institutions claim to find an irreducible need for numerical complexity.

    So what’s all this you say about there being no analogy between complex numbers and God?

  7. #7 Richard Wein
    September 5, 2007

    It doesn’t make much sense to me to discuss whether imaginary numbers exist. Mathematical concepts are mere abstractions, and mathematicians can invent any abstractions they like. Some abstractions, like natural numbers, can be mapped onto reality in a simple way. Others may have a much less direct connection with reality (or perhaps none at all).

    On the other hand, the question of God’s existence is a question about reality. He either exists or he doesn’t (at least once you’ve defined what you mean by “God”).

    To compare God with imaginary numbers in the way that Shaw does makes little sense.

  8. #8 csrster
    September 5, 2007

    Quite. If she’d drawn an analogy from science, as opposed to maths, then there would be something there to refute. As it is, her argument isn’t even wrong.

  9. #9 Sam
    September 5, 2007

    Irrational, imaginary, got it. Now perhaps the religious folk could develop a theory of how we might do complex arithmetic using “delusional numbers”? Could be fun!

  10. #10 Bob Carroll
    September 5, 2007

    Historically, mathematicians had a difficult time accepting zero as a number as well. Maybe God is analagous to zero?

  11. #11 Guitar Eddie
    September 5, 2007

    I never cease to be amazed at the lengths some religionists will go to make their beliefs seem rational to the rest of the world. Why bother? Why waste all that energy trying to prove something you know can never empirically demonstrated?

    I would rather just say that there are things I believe. I know they’re not rational or scientifically demonstrable, but they make me happy and functional in this world.

    If religious people would adopt the above attitude, they would save themselves needless conflicts and a lifetime’s worth of pointless grief.

    GE

  12. #12 bill
    September 5, 2007

    Ohh – Does that mean that god of her god is a quaternion, and its god is an octonion? Meta-gods! This is great! But do they commute?

  13. #13 Tegumai Bopsulai, FCD
    September 5, 2007

    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.

    No! Do not stare into the sun:

    Blinded by Faith
    June 19 2007 at 04:24AM
    By Alex Eliseev
    Amal Nassif believed 17-year-old Francesca Zackey when she said the Virgin Mary would appear if Nassif gazed into the sun.
    Now 37-year-old Nassif, a devout Catholic, may be blind for life.
    …

  14. #14 Jason Rosenhouse
    September 5, 2007

    Richard-

    In fairness to Ms. Shaw, she acknowledges that the precise sense in which numbers can be said to exist, if at all, is a thorny problem. I don’t think that issue really affects her argument, such as it is.

    As for whether numbers exist, I tend to agree with the point of view of Timothy Gowers. Gowers is a Fields Medalist who wrote a little book called Mathematics: A Very Short Introduction. I don’t have the book handy, so I can’t look up the exact quote. But at one point he mentions that philosophers certainly spend a lot of time worrying about whether numbers exist. Then he said something like, “In this they differ from mathematicians, who either regard it as obvious that numbers exist or else feel they don’t understand what is being asked.”

  15. #15 Blake Stacey
    September 5, 2007

    Excellent post, Jason.

    For the fun of it, I’d like to try multiplying Shaw by -1.

    Religious mystics, pencil and paper in hand, peer through stained glass 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 the ancient myths of our ancestors call heretical.

    Migosh, she’s -1 times right!

  16. #16 Thony C.
    September 5, 2007

    Shaw presents a passable history of imaginary numbers.

    In this you are right and I was quite surprised at how passable it is, I have read worse from mathematitions. However her presentation of the poor Cardano is total bullshit and her grasp of Renaissance science is utterly miserable.

  17. #17 Fred
    September 5, 2007

    Shaw has won me over: God is imaginary.

  18. #18 mark
    September 5, 2007

    Scientific positivists, pencil and paper in hand, peer through shatterproof, UV-protected glasses at a world of animals, vegetables, and minerals. But genuine religionists–true seekers of knowledge–are not afraid to let the sunlight dazzle them. They can see Circles, and Colors, Unicorns, and…even God.

  19. #19 Blake Stacey
    September 5, 2007

    Whew! I meant to write a brief note acknowledging this post, and it grew into a whole post of its own.

  20. #20 bigTom
    September 5, 2007

    I rather think the analogy is somewhat appropriate.
    Imaginary numbers have no physical existance, but the concept affects the may many of us think. Now substitute God for imaginary numbers, and the statement is still true. The concept provides a framework for the way people understand the world. Of course debating about whether imaginary numbers exist is kind of pointless.

    The other point we hear from religious scholars, is that faith is meant to be faith (i.e. something one believes without proof), so trying to prove it is really missing the whole point.

  21. #21 Loren Petrich
    September 6, 2007

    The problem here is that numbers are NOT physical objects — they are abstractions. The metaphysical status of abstractions is a rather knotty question that I will not go into here, but it can be shown that imaginary numbers are as “real” as any other kind of numbers. In fact, “imaginary number” is a rather misleading term, just as “negative number” is.

  22. #22 SLC
    September 6, 2007

    One waits with bated breath for the IDiots to proclaim that dark energy proves the existence of god.

  23. #23 Joseph Hertzlinger
    September 6, 2007

    I thought higher set theory was the branch of mathematics usually described as “exact theology.”

    So if you’re losing your faith in the Axiom of Choice…

  24. #24 Blake Stacey
    September 7, 2007

    Tyler DiPietro’s rebuttal is much shorter than mine.

  25. #25 Anonymous
    September 7, 2007

    If a quadratic equation has real-number solutions, its curve will intersect the X-axis at those numbers, but if the solutions are imaginary numbers, the curve will not intersect the X-axis at all, so in that sense the imaginary numbers are really imaginary. However, imaginary-number mathematics has been found to correspond to certain physical phenonema, like aerodynamics and the behavior of alternating current circuits. For example, in the Joukowski transformation of conformal mapping, the aerodynamics of a rotating cylinder is used to determine the aerodynamics of fixed-wing airfoils! See — http://www.grc.nasa.gov/WWW/K-12/airplane/map.html

  26. #26 Robert O'Brien
    September 8, 2007

    And not to put too fine a point on it, but it is worth mentioning that all of the 15th. century mathematicians that initially scoffed at imaginary numbers were Christian theists, not stodgy atheists.

    Uh-huh. And how many of those advancing complex analysis were atheists? Euler? No. Cauchy? No. Gauss? No.

  27. #27 Tyler DiPietro
    September 8, 2007

    “Uh-huh. And how many of those advancing complex analysis were atheists? Euler? No. Cauchy? No. Gauss? No.”

    And thus Robert O’Brien concludes yet another irrelevant temper tantrum.

    My point wasn’t that the complex numbers debacle demonstrated the superiority of atheism, it was only the irony of Shaw attempting to insinuate the superiority of theism by using a scenario where the antagonists were uniformly theists.

  28. #28 Christophe Thill
    September 12, 2007

    And what about irrational numbers? Aren’t they… well, irrational? Just like religious faith is said to be! QED.

    It looks like we’ll always be victims of the power of words. Because the idea went against common sense (and current mathematics, because people still thought that mathematics should not contradict common sense), “imaginary” numbers received that silly name. When you think about it, all numbers are imaginary, including the naturals. You can give me three apples, but you can’t give me three. Three is a concept.

    What if we found a new name? Why not rename imaginary numbers? Preferably one witn no old meaning attached. Something like “bormo” (for “based on the root of minus one”). Is that too silly?

  29. #29 Martin
    September 12, 2007

    Paraphrasing Kierkegaard, faith exists where rational thought ends. One who does not doubt, does not have faith. Theists don’t live without doubt, and most recognize that the existence of God is not belief that exists in logical or rational thought. Theists have just resigned themselves that doubts will always persist and rational thought is constraining. Highly rational and logical people still make irrational and illogical choices in their lives, is it simply that we are all cognitively disabled or is life bigger than the constraints we apply?

  30. #30 Martin
    September 12, 2007

    Paraphrasing Kierkegaard, faith exists where rational thought ends. One who does not doubt, does not have faith. Theists don’t live without doubt, and most recognize that the existence of God is not belief that exists in logical or rational thought. Theists have just resigned themselves that doubts will always persist and rational thought is constraining. Highly rational and logical people still make irrational and illogical choices in their lives, is it simply that we are all cognitively disabled or is life bigger than the constraints we apply?

  31. #31 Anonymous
    September 14, 2007

    Christophe Thill | September 12, 2007 5:02 AM said,

    And what about irrational numbers? Aren’t they… well, irrational? Just like religious faith is said to be!

    I think that the root for “irrational” in “irrational number” comes from “ratio” rather than from “rational” (meaning reason). Irrational numbers are numbers that cannot be exactly expressed as ratios of whole numbers. The reason for that is fairly apparent — irrational numbers are infinite series of fractions, and therefore can never be expressed exactly.

    Because the idea went against common sense (and current mathematics, because people still thought that mathematics should not contradict common sense), “imaginary” numbers received that silly name

    As I pointed out, an “imaginary number” is a polynomial’s root that is not a point of intersection of the polynomial’s curve and the X-axis — and in that sense an imaginary number is truly imaginary.