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The other day, I noticed someone with the equation ei π = -1 as their usericon. This equation, or rather the high esteem in which it is held by so many, infuriates me, and I'm going to try to explain why.
Actually, I have nothing much against the equation itself. It has a pleasant - well, not exactly symmetry, but you know what I mean. What annoys me is people who (in many cases, literally) think it's the final proof of the existence of God. It isn't. It's a slightly more sophisticated cousin of those "take away the number you first thought of, multiply by seven, and add the digits together; now, is your answer sixty-nine?" trivial results from Christmas crackers.
Here's why: it's a tautology. Not just in the sense that every mathematical theorem is a tautology, but in the sense that it's actually a very shallow result, following almost immediately from the definitions involved. Just think, for a minute, about how you'd define the numbers mentioned. π first, because that's the hard one. "The ratio of the circumference to the diameter of the circle" - OK, but that's pretty fuzzy: what does it mean? Well, the line integral around a circular path, divided by twice its radius. Do some more simplifications, and you'll end up with the following: "π is the smallest positive real solution to the equation sin x = 0". Now, how do we define sin? Since we're working with complex numbers, we're pretty much forced to use the power series definition (which acceptable rigour more-or-less forces us to use anyway). The result that exp(z) = cos(z) + i sin(z) [1] drops straight out, as does the result that cos^2(z) + sin^2(z) = 1 [2] (OK, the last one's not that straightforward).
Apply this to the term exp(iπ). From [1], we know that exp(iπ) = cos(π) + i sin(π), and by definition sin(π) = 0. From [2], cos^2(π) = 1; since π is, by definition real, so must cos(π) be, since the power series has only real coefficients. So cos(π) = +/- 1, hence exp(iπ) = +/-1. We're not all the way there yet, but note that the equation ei π = 1 would have all the acclaimed aesthetic properties of the true equation.
So, what may initially have appeared as a deep and mysterious result is revealed as nothing of the kind. Note that the only non-obvious fact I've used is the definition of π as a solution to sin x = 0, and if you think about it for a minute that's actually perfectly natural. And yet this result gets parroted everywhere, in spite of the fact that there are plenty of genuinely surprising and deep results out there. The insolubility of the general quintic. Yoneda's Lemma. The Four-Colour Theorem. Stuff like that. Edit: since we're talking about complex analysis, how about Cauchy's theorem, or better, the Residue Theorem? That blows me away. Or the other Cauchy's theorem, the one about groups?
As for "final proof of the existence of God" - no, no it isn't. Mathematical truths cannot prove God's existence. Quite simply, God has no say in the matter of whether ei π is 1 or -1 (assuming, for the sake of argument, that He exists). If politics is the art of the possible, then mathematics is the art of the necessary. Mathematical truths simply could not be any other way. They do not depend on the existence of the Universe for their truth. This infuriated me when I read the end of Carl Sagan's Contact: God couldn't have left His final message to his creation in the digits of π, because God had no say in what those digits would be. If He was going to leave us such a message, he'd have to do so in the digits of some appropriate physical constant (a dimensionless one, probably).
[Actually, there's a lovely story about a "mathematical proof of God's existence" given by Euler himself, recounted here.]
[Edit: in a few of my comments below, I've confused de Moivre's theorem (cis(nθ) = cis(&theta)^n) with Euler's formula (equation [1] above). Sorry for any confusion.]
[Further edit:![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
foreverdirt, whose userpic prompted this rant, has graciously replied:
Actually, I have nothing much against the equation itself. It has a pleasant - well, not exactly symmetry, but you know what I mean. What annoys me is people who (in many cases, literally) think it's the final proof of the existence of God. It isn't. It's a slightly more sophisticated cousin of those "take away the number you first thought of, multiply by seven, and add the digits together; now, is your answer sixty-nine?" trivial results from Christmas crackers.
Here's why: it's a tautology. Not just in the sense that every mathematical theorem is a tautology, but in the sense that it's actually a very shallow result, following almost immediately from the definitions involved. Just think, for a minute, about how you'd define the numbers mentioned. π first, because that's the hard one. "The ratio of the circumference to the diameter of the circle" - OK, but that's pretty fuzzy: what does it mean? Well, the line integral around a circular path, divided by twice its radius. Do some more simplifications, and you'll end up with the following: "π is the smallest positive real solution to the equation sin x = 0". Now, how do we define sin? Since we're working with complex numbers, we're pretty much forced to use the power series definition (which acceptable rigour more-or-less forces us to use anyway). The result that exp(z) = cos(z) + i sin(z) [1] drops straight out, as does the result that cos^2(z) + sin^2(z) = 1 [2] (OK, the last one's not that straightforward).
Apply this to the term exp(iπ). From [1], we know that exp(iπ) = cos(π) + i sin(π), and by definition sin(π) = 0. From [2], cos^2(π) = 1; since π is, by definition real, so must cos(π) be, since the power series has only real coefficients. So cos(π) = +/- 1, hence exp(iπ) = +/-1. We're not all the way there yet, but note that the equation ei π = 1 would have all the acclaimed aesthetic properties of the true equation.
So, what may initially have appeared as a deep and mysterious result is revealed as nothing of the kind. Note that the only non-obvious fact I've used is the definition of π as a solution to sin x = 0, and if you think about it for a minute that's actually perfectly natural. And yet this result gets parroted everywhere, in spite of the fact that there are plenty of genuinely surprising and deep results out there. The insolubility of the general quintic. Yoneda's Lemma. The Four-Colour Theorem. Stuff like that. Edit: since we're talking about complex analysis, how about Cauchy's theorem, or better, the Residue Theorem? That blows me away. Or the other Cauchy's theorem, the one about groups?
As for "final proof of the existence of God" - no, no it isn't. Mathematical truths cannot prove God's existence. Quite simply, God has no say in the matter of whether ei π is 1 or -1 (assuming, for the sake of argument, that He exists). If politics is the art of the possible, then mathematics is the art of the necessary. Mathematical truths simply could not be any other way. They do not depend on the existence of the Universe for their truth. This infuriated me when I read the end of Carl Sagan's Contact: God couldn't have left His final message to his creation in the digits of π, because God had no say in what those digits would be. If He was going to leave us such a message, he'd have to do so in the digits of some appropriate physical constant (a dimensionless one, probably).
[Actually, there's a lovely story about a "mathematical proof of God's existence" given by Euler himself, recounted here.]
[Edit: in a few of my comments below, I've confused de Moivre's theorem (cis(nθ) = cis(&theta)^n) with Euler's formula (equation [1] above). Sorry for any confusion.]
[Further edit:
foreverdirt, whose userpic prompted this rant, has graciously replied:That it falls out so vacuously from the definitions is one of my favourite things about the equation, but I'm sorry the emphasis some people put on it irritates you.How entertainingly ironic :-)]
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