Skeptophilia (skep-to-fil-i-a) (n.) - the love of logical thought, skepticism, and thinking critically. Being an exploration of the applications of skeptical thinking to the world at large, with periodic excursions into linguistics, music, politics, cryptozoology, and why people keep seeing the face of Jesus on grilled cheese sandwiches.

Saturday, January 18, 2014

Euler's identity, and seeing the divine in mathematics

Yesterday I ran into a "proof of the existence of god" I'd never seen before; the idea that there are mathematical patterns that suggest the hand of a deity.

One of the most popular patterns that religiously-inclined mathematicians point to is "Euler's identity:"

And on the surface of it, it does seem kind of odd.  "e" is the base of the natural logarithms; "i," the square root of -1, and thus the fundamental unit of imaginary numbers; pi, the ratio between the circumference of a circle and its diameter.  That they exist in this relationship is certainly non-intuitive, and the non-intuitive often makes us sit back, and go, "Wow."

Euler's identity isn't the only such set of patterns, though.  A gentleman named Vasilios Gardiakos goes through a good many mathematical gyrations to show that god wrote his signature in number patterns, including the presence of "Pythagorean triplets" in the decimal expansion of pi.  (A "Pythagorean triplet" is a set of three integers that solve the Pythagorean theorem, that the sum of the squares of the two sides of a right triangle is equal to the square of the hypotenuse.  The most famous one is 3, 4, and 5.)

Gardiakos's messing about seems to me to stray a little too close to numerology for my comfort.  If you've already have decided that number patterns Mean Something, and you're willing to use any pattern you find, you're already off to a good start.  Add to that the fact that he was searching for patterns in decimal expansions that are infinite (pi, e, and √2), and it's a sure bet that given enough time, you'll come across whatever you need.

The use of the Euler identity, though, is a little harder to answer.  It certainly seems... well, perfect.  It relates five fundamental constants in mathematics -- e, pi, i, 1 and 0 -- in one simple, elegant equation.  And the mathematicians themselves have waxed rhapsodic over it.  Mathematician and writer Paul Nahin calls it "the gold standard for mathematical beauty."  Mathematician Keith Devlin of Stanford University states, "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence."

Which is all well and good, but does it prove anything beyond a fascinating and complex mathematical relationship?  First of all, the fact that it's true might be non-intuitive, but it is hardly a coincidence.  For a lucid explanation of why Euler's identity works, you have to go no further than the Wikipedia page on the subject, which leads us step-by-step through a proof of how it was constructed.

And honestly, all of the theologizing over beautiful theorems in mathematics seems to me to turn on one rather awkward question; if you are claiming that Euler's identity, or any other mathematical pattern, proves the existence of god, you are implying that had god wanted, he could have made the math work differently.  God exists -- we get Euler's identity and various patterns of numbers in the decimal expansion of pi.  God doesn't exist -- we don't.

So then, can you conceive of a mathematical system in which Euler's identity is a false statement?  Because if not, then god (should he exist) was apparently constrained to creating a universe where Euler's identity was true, and the god/no god models end up looking exactly the same.

Kind of a poor proof, honestly.

What this sort of thing seems like, to me, is an extension of the Argument from Incredulity: "I don't really understand how this could be true, so it must be god."  Understanding Euler's identity does require that you know a good bit of mathematics; easier, maybe, just to marvel at its beauty, and attribute that beauty to a deity.

For me, I'd rather just try to understand the reality, which is marvelous enough as it is, and worth reveling in a little.  It might be time to break out Douglas Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid and K. C. Cole's The Universe and the Teacup again.


  1. I remember when we walked through this proof in one of my calculus lectures. Its beauty is undeniable and pure.

    Later that year, however, a few of us proved that god is a simple harmonic oscillator that travels in the imaginary plane that crosses over the real plane once every 2000 years. So there's that.

  2. Can Euler's identity equal precisely 0 anyway, considering e, i and pi are irrational?

    1. Yes. The limit of a sequence as N _goes to_ infinity is the number (assuming it is a convergent sequence) the sequence tends to. _At_ infinity the limit IS the given number.

  3. ...and what of, once again, humanizing God, should it exist?

    People like beautiful things, God likes what we like, and therefore God finds specific mathematical equations beautiful because we do, and created them for this reason.

    It takes a lot of hubris to think we know the "mind" of God through our literal experiences and observations.

  4. Euler thought this equation explained the moment of creation. You might even call it a mathematical, first mover argument. The equation explains the spiral movement of time and quite literally the beginning of time.

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