Euler's Equation

There are certain things in life which strike people as proof of existence of a transcendent power, a mystical presence of something beyond the mundane laws of the sciences. To some, Euler’s equation connecting i, 1, 0, pi and e is one such divine object, a signal set by the Creator to pronounce, “I AM”. The author takes this idea and runs with it…

So one day, Euler’s Equation works, and then, it doesnt. Mathematics has changed, somehow. Prof Carlton remarks:

“Euler’s equation was the proof,” I said. “It was God’s covenant with mankind, like the rainbow after the flood. It said, the universe isn’t just chance, it’s designed. All the basic elements of reality fit together like a jigsaw. The equation is God’s signature, just to let us know he’s still here. But now the equation doesn’t hold; God has left the building.”

People are perplexed, but people don’t know what it is that is changing. For the next day, imaginary numbers stop existing. Numbers start failing, as does Geometry.

“The universe is getting simpler. We’re winding down. The game’s over,” I said. “It’s as if the exam’s just finished, and we’ve handed in our papers. You can carry on working through the problem if you want, but it won’t affect your grade any more. It’s too late to be good..”

And then… philosophical speculations which I found to be very appealing hand-waving and interesting reading, if too short. Is this God’s handiwork? Is Euler’s equation “complete” (I suppose “modifiable” to form a new mathematical universe)? If it is not, does that point to a path to the higher reality orthogonal to the one which we have assumed? As the conclusion goes,

“We’re working to find the equations that describe the real universe. Somewhere, there is a world with a God, where real people can work and love. Where Euler’s equation [Reviewer: the left hand side of Euler’s equation] doesn’t make 0. And this time, we (mathematicians) won’t just describe it. People look to us for answers now, and we will find them. With our constants and our mathematical relationships, we will find God.”

Of course, the idea that Euler’s Equation changes, along with many other things but leaves large parts of the working universe intact during the gradual wind-down process challenges the fact that all of mathematics is interconnected, flowing for the most part from some basic set-theoretic notions (ZFC). However, once you invoke God and a higher-dimensional shadow, it is easy to believe that the mathematical changes envisioned form a smooth(ish) eversion process, where the axioms are somehow changing consistently to ensure that Euler’s equation and many other things break down gradually but other implications of the original ZFC continue to be respected, in a Reverse Mathematics style. In any case, demanding such consistency of fabulist or absurdist fiction is folly :-)