Valentine Confession

Valentine’s day is upon us, so I think it’s a good time to say this:
I love mathematics.
I suppose that a large part of that is similar to what some people look for in their beloved ones: the mystery, surprise. Mathematics, with all her hard-set iffs and irrevocable deduction, always manages to surprise me. From simple axioms, so straightforward they are barely worth mentioning, come results that leave the reader dumbstruck. I have had several of these moments over the past semester. Most of them contain a mixture of awe, disbelief, and excessive cursing.
It can be shown, for example, that the following identity exists:

\int_{-\infty}^{\infty} e ^{-x^2}dx = \sqrt{\pi}

“What? What?“
“But why?“
“What does \pi have to do with e? Why should those be connected in that way?“
“What does the area under the curve of an exponentially decaying function, which is defined as the limit of some strange series, be equal to the square root of the ratio between the circumference and diameter of a circle in the plane?”

Of course, I’m not talking about the proofs, such as the one referenced. That is technical, and understood. But all sorts of relations keep popping up in places where you don’t expect them. Now, it is true, \pi is many things, besides the ratio between the circumference and diameter of a circle in the plane. And e also appears in many places. Perhaps they are linked somewhere else. Maybe that can be shown. But I wonder, if there is an intuitive, non-formal way, that will satisfy that inner voice which constantly whispers, “how are those two connected?” At the present state, I understand how to solve the integral, but I cannot say I understand where the answer comes from.
And this is just one example, out of many. Mathematics’ ability to say something definite and indisputable about the world, and yet something so inexplicable, is marvellous.
And so, this Valentine’s day is dedicated to you, mathematics.
(You’re the only one who would date me, anyway)

3 thoughts on “Valentine Confession

  1. Also, there are N people who would date you, assuming an infinite population (Earth is close enough to that). I’d challenge you to prove that all will disqualify you. This would require proving that for each and every person on Earth, you’d have a disqualifying trait.

  2. Just a small remark (i know it’s not the point of the post…): The way I like looking at this identity is as a specialization of the fact that the Gaussian is its own Fourier transform (up to a constant related to pi). You plug 0 in the Fourier transform and you get the identity.

    Maths is awesome.

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