Jim Papa emailed me with an improvement on my December 14 story of Euler proving the existence of God by formula. He proposes a different formula which makes sense of the story, writing that he thinks the equation of Euler that ought to be in the story is

That equation is one of several "Euler's Formulas out there (Euler did a lot of fundamental work) and probably the one that best deserves the name. (Another one-- the one about adding up vertices-- is the one used in Lakatos's Proofs and Refutations.) It is, to be sure, not a proof of the existence of God, but it at least has some relevance to the issue. Each of the symbols in it is a natural constant of huge importance. If you were asked for the five most important numbers, these are probably what a mathematician would tell you, for good reasons. The number e is the only number such that the slope of the function f(x) = e^x equals e ^x. The number i is the square root of -1, the basis of the imaginary numbers. The number &pi is the ratio of the circumference of a circle to its diameter. The number 1 is the only number which doesn't change the value of any other number multiplied by it, and is also the difference between adjacent integers. The number 0 is the only number which doesn't change the value of any other number to which it is added, and the only number which when multiplied by a different number yield itself. Note, too, that the equation contains the concepts of multiplication, subtraction, exponentiation, and equation. So there is a lot going on in a small space.

As a result, the equation is amazing. What a coincidence that everything links together so neatly! This reminds me of the Great Watchmaker or Intelligent Design ideas that everything in Nature links together so neatly that there must be a Creator. It is different, though, in an important way. It may be logically impossible to construct a satisfactory mathematical or logical system in which Euler's formula is not true. But there are lots of unsatisfactory systems, perhaps, in which it is not. Those systems just don't work very well for any practical purpose. Human construction both more and less than Nature. More, because it is a system we have written down to fit our particular way of thinking. Less, because at least any other beings who think like us, even if they do not sense like us, would come up with the same mathematics. But here I am in too deep water for someone inexpert in mathematics and short on time. What I have just said may well be nonsense.

At any rate, to explain what Jim says above, the numbers e and &pi are transcendental, i is imaginary, 0 is nothing, and 1 is the humble building block for the integers and rational numbers, and, eventually, everything else. An allegorist could go to town on this. As a start, let 1 be Man, 0 be Evil (Augustine thought of Evil as Nothingness; in Faust Mephistopheles calls himself the spirit that ever says no), and I'll leave the rest to you.

Note that Euler's Formula links to yet another mathematical idea, that of the sine function from trigonometry, via the formula, and setting x = π.

For a little more, go to the Euler's Theorem page at the Famous Problems and Proofs website. Also, Alex L�pez-Ortiz of U. of Waterloo has interesting things to say about defining exponentiation, and Euler's Theorem. This kind of math writing is not too hard in HTML. Use the VIEW SOURCE command in your browser and search for

EXPRESSION 1: e^iπ

EXPRESSION 2 (not italicized): e^iπ

EXPRESSION 3 (not italicized, different spacing): e ^{i π}

EXPRESSION 4 (just the Greek letter): π

EXPRESSION 5 (just superscript and subscript): eⁱ + e_i

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