I once argued with a woman online — can you imagine? Me, arguing online! — Who claimed that global warming couldn’t possibly be due to human activity, because of the small amount of CO2 our activities release compared to the total in the atmosphere. So I slipped her 500 µg of LSD, and said “Let’s see what small amounts of a chemical can really do!” Ha ha! It was online, so I could not do that. But imagine if she had never encountered ice but only water between 100°F and 32.5°F. I’m sure if I tried to explain to her that the next drop of 1° would make a huge difference, she would scoff, and say “No, the water is just going to get a little more dense and a little more sluggish.” Dynamical systems experience phase transitions, where a small move past some point throws the system into a whole new form of behavior.
Either I'm missing a pun, God is less impressive than I thought, or you are more easily impressed than I thought.ReplyDelete
Or maybe you are just not that sharp at mathematics.Delete
The physicist Richard Feynman called the [above] equation "our jewel" and "one of the most remarkable, almost astounding, formulas in all of mathematics."
Math Wiki says, "When Euler's formula is evaluated at θ = π, it yields the simpler, but equally astonishing Euler's identity."
I can keep going if you want, finding you mathematician after mathematician describing how "astonishing" and "amazing" this identity is. Don't worry: we can't all have an appreciation for all subjects!
This formula relates the five most fundamental numbers in mathematics in a formula simpler than anyone could have imagined existed before Euler discovered it.Delete
And, of course, if we have the expanded version:
cos θ + i sin θ = e^(iθ)
We have elegantly related the exponential and trigonometric functions, in a way no one expected could be done before Euler!
Top that all off with the fact that this formula emerges naturally in the process of solving second-order differential equations, and I think we pretty much have a "God trifecta" here.
"The physicist Richard Feynman called the [above] equation..."Delete
He was referring to Euler's formula, not Euler's identity.
"...we have the expanded version:"
No, Euler's identity is a special case of Euler's formula (actually one of two special cases where the imaginary part is zero. There are also two special cases where the real part is zero.)
The interesting thing to me about Euler's identity is that it makes it clear that Pi being defined as the ratio of a circle's circumference to its diameter was the wrong choice. If Pi was defined as the ratio of a circle's circumference to its radius Euler's identity would be e^pi*i = 1. Now *that's* beautiful. As it is written in your post, it's precisely 180 degrees from perfection. Evidence of the devil perhaps?
Greego, you are new here, so welcome, and let me tell you why in the future I won't bother posting comments from you like the above:Delete
1) "He was referring to Euler's formula, not Euler's identity."
You really think that he wouldn't have meant his remarks to apply to both, since, as someone points out, one is just a special case of the other? I sure do.
2) "No, Euler's identity is a special case of Euler's formula..."
Oh boy, now I'm really thinking "delete." What is the general case in relation to the special case? Well, it is an EXPANDED version of it, isn't it, applying more generally! So you just told me "No," and then re-stated exactly what I said in different words.
"The interesting thing to me about Euler's identity is that it makes it clear that Pi being defined as the ratio of a circle's circumference to its diameter was the wrong choice."
The interesting thing about this remark is it shows an inability to see through arbitrary choices of symbols to the underlying mathematical truth, which is exactly the same whether pi is defined in terms of the radius or diameter, and certainly not 180 degrees from the same, as you suggest.
The real point Daniel is that naively these seem to have nothing to do with each other. Pi is a ratio of a circles circumference to its diameter. e is the infinite limit of a certain formula in calculus that seems unconnected to circles. i is the 'imaginary number' that no-one could understand but seem to help solve cubic equations. That's reall all that was known about them before Euler. And then suddenly this formula connects them all, an simply. It's a real shock the first time you see it if you actually know all the other stuff.Delete
Of course the "proof" of god is wrong.
"Proof of" was hyperbole: obvious, I hoped. "But really good argument for" is quite accurate: it is just this sort of thing that made theists out of Pythagoras, Plato, Aristotle, and in modern times Kurt Godel. And Euler himself was of course deeply religious, and certainly would have viewed his own work as strong evidence of the divine.Delete
Thank you for dropping by Greego. See ya around the Internet sometime.Delete
Good post, Gene.ReplyDelete
Thanks. I think even an atheist like Feynman, who actually understood the significance of the above formula, who have said, "Yeah, I certainly can see how someone might take this as evidence for God!"Delete
Unfortunately I can't remember ever reading him say anything like that. He was pretty upset when some orthodox Jewish guys asked him if electricity were fire because they wanted to know if they could use their lights on a holy day. (I might be botching the details.) And there's a video floating around of him saying, "I'm not afraid to admit I don't know," where he's criticizing religious people.Delete
I'm not saying your observation here is wrong, just that unfortunately, the atheists have a pretty good claim on Feynman.
We need a new word then Gene: Eulier-than-thou.ReplyDelete
My complex analysis prof, when he proved it, looked at the class for reactions. Most of the class was applied math types (pure math was a small cohort at my school) who just took it in stride: oh, another formula. Those few of us who were stunned he said were the real mathematicians.ReplyDelete
You mean the real theologians. The prof must have had a thick accent.Delete
He sounded exactly like Steve Landsburg in fact!Delete
I think you need to be more explicit here at the end...ReplyDelete
An old maths teacher once asked us what we thought was the most interesting number in that identity. He seemed offended when I said i...ReplyDelete
This isn't really an argument for God so much as a matter of interpretation of the formula.ReplyDelete
You can rearrange Euler's identity in an interesting way. First, consider the following definition:
tau = ratio of the circumference of a circle to its radius, or 2*pi.
Note that the definition of a radian angle measure is "theta = arc length formed by the angle theta applied to a unit circle / radius of the circle", and that tau radians corresponds to 360 degrees or one complete rotation around a circle.
You can use the memetic "-T-au radians equals one -T-urn around a circle" to remember this. Arguably, Tau is the more fundamental circle constant, relating the length of radius (which is one half of the definition of a circle, the other being its origin point) and the circumference, and making for a less confusing unit circle diagram.
Now, onto Euler's identity. It involves complex exponentiation, which is deeply connected both to the circle functions and to the geometry of the circle itself by the very definition of a unit circle [see note 1 below].
Consider the following rearrangement of Euler's identity:
e^(i*tau) = 1
Stated simply, it implies that the complex exponential of the fundamental circle constant is the multiplicative identity, 1.
Geometrically speaking, multiplying x by e^(i*theta) corresponds to rotating the complex number x by an angle theta in the complex plane.
This suggests a second interpretation of Euler's identity:
A rotation by one full turn around an origin point brings you back to an equivalent position (holding true in the complex plane and for all complex-plane rotations as well).
Euler's identity, when properly posited via the use of Tau rather than Pi, is simply true by definition.
While it is indeed miraculous that these numbers, which appeared to be entirely separate developments of mathematics, are actually quite interrelated, his identity should not be considered an argument in favor of God.
(I take no credit for the above work, having been paraphrased and adapted from tauday.com/tau-manifesto; I did, however, extrapolate the conclusions to suit the discussion at hand)
[note 1]Some people might claim it’s because Euler's identity relates them, but you can get an intuitive understanding of why they’re related, and then use that to formulate Euler’s identity, or “explain” why it holds true.
"While it is indeed miraculous that these numbers, which appeared to be entirely separate developments of mathematics, are actually quite interrelated, his identity should not be considered an argument in favor of God."Delete
No, no: nothing miraculous must EVER be considered an argument in favor of God! Because he is the "sky bully!" Listen to Dr. Zoiberg!!!