Beauty in Mathematics: Exploring Aesthetic Beauty in Proofs & Papers

[Ackbach]

This is a bit of an odd problem - more philosophical, perhaps. I've been reading The Art of Mathematics, by Jerry P. King, and the author points out that mathematicians do mathematics for aesthetic reasons - for beauty. They see beauty in the mathematics that they do. My question is this: what is the nature of this beauty? What constitutes a "beautiful proof"? What is a "beautiful paper"? It seems to me that many mathematicians intuitively know that a result is beautiful, but have a hard time explaining why it's beautiful.

A few of my thoughts:

1. The medievals thought of beauty as "that which has form, harmony, and complexity". While this seems a great definition for the arts, I'm not so sure it's adequate for mathematics, though it might have some relevance. What's your opinion?

2. If a theorem assumes little and proves a lot (assuming it's valid!), that's usually considered beautiful, right?

3. If there's a clever construction in the proof, that's considered beautiful, right?

4. Is "elegant" a synonym for "beautiful" in mathematics? Baby Rudin has many "elegant" approaches to proofs, supposedly, but

5. Is the "elegant" approach necessarily the most pedagogical?

6. How can we help students see the beauty in mathematics?

There's a relevant question on Math.SE here. There is, interestingly, a wiki on this as well.

 

[Deveno]

My college friend read somewhere a "formula" for beauty:

Beauty = Order/Complexity.

As with most formulae trying to capture the ineffable, it falls short in many respects-a simple mechanical system should thus possesses great beauty, but this is not what we mean. But it does point out one important aspect-beauty in mathematics should have a sense of "proportion", something ought not to be *needlessly* complicated.

Elegance in mathematics is "one" yardstick, but there are pitfalls here, too-often, one seeks a certain terseness to exhibit elegance, and this can hinder the beauty inherent in a result, because there is too much left unsaid.

Clarity is another-and yet, too much clarity robs us of some of the joy of discovery that is one of the secret pleasures of mathematics. Stating the obvious, no matter how profound, does not quite qualify.

There are some heuristic guidelines: results which possesses "duality", a certain symmetry of exposition, are often more aesthetically satisfying than a chain of restrictive implications. It helps if something is somewhat surprising, as well-Ivan Niven's proof of the irrationality of pi comes to mind: the result appears to "come out of nowhere" without even assuming much about pi itself.

 

[ModusPonens]

Picking up on Deveno's formula, I think, as mathematical geeks, we could try to improve the formula. My attempt is

\(\displaystyle Beauty = \frac{Neatness \times Synthesis \times Influence \times Surprise}{Length \times Dryness} + (Mystery)^2\)

Where,
- Neatness is, well, neatness of the proposition.
- Synthesis is the power to bring a lot of different things together.
- Influence is the degree to which the theorem has numerous implications to other theorems, etc.
- Surprise is the surprising factor of a theorem.
- Length is the length of the wording and conditions for the theorem to be true.
- Dryness is the level of absence of palpable intuition assossiated with it.
- Mystery is the the degree to which we don't understand why the theorem is true and it's different from surprise. It is squared because when a result is obvious, it's not very beautiful (x < 1 implies x^2 is even smaller), but when it is intriguing, it has a high value. And since this is a bonus feature that is less clear, I added it instead of multiplying it.

Improovements are welcome. I expect subjectivity to kick in as well. ;)

 

[Ackbach]

Thank you for all the replies; a formula could be helpful, but I doubt anyone formula would capture all the nuances of beauty and elegance in mathematics. Here are the last few closing paragraphs of The Art of Mathematics, which literally had me in tears when I read it. This gets at it a bit:

The gunfighter always found another town to tame. So far, I have always found another course to teach. But the Vermont summer passed long ago. Courses are running out. Soon I will be down to just one.

Just one more course and I'm done. Make it classical complex variables. Let me do it once more.

One day when the wind is right I'll do the Cauchy Integral Formula for the last time and I will do it truly. I will write it carefully and the students will see the curve and the thing inside and the lazy integral which makes the function value appear as quickly as my palm when I open my hand.

They will see the art of mathematics. And they will never care for anything half as much.

 

[Deveno]

One of my favorites:

Let $f: [-1,1] \to \Bbb R$ be the (rectifiable, since $f$ is continuously differentiable on its domain) function:

$f(t) = \sqrt{1 - t^2}$. We have $f'(t) = \dfrac{-t}{\sqrt{1 - t^2}}$.

We have the arc-length of $f$ as:

$$\int_{-1}^1 \sqrt{1 + [f'(t)]^2}\ dt$$

$$= \int_{-1}^1\sqrt{1 + \dfrac{t^2}{1 - t^2}}\ dt$$

$$= \int_{-1}^1 \sqrt{\dfrac{1-t^2 + t^2}{1 - t^2}}\ dt$$

$$= \int_{-1}^1 \dfrac{dt}{\sqrt{1-t^2}}$$

If we employ a change of variable, by setting: $t = \sin u$, so that $dt = \cos u\ du$, we see that our limits of integration change from $-1$ and $1$, to $\arcsin(-1)$ and $\arcsin(1)$, and our integral becomes:

$$= \int_{\arcsin(-1)}^{\arcsin(1)} \dfrac{\cos u}{\sqrt{1 - \sin^2u}}\ du$$

$$= \int_{-\pi/2}^{\pi/2} \dfrac{\cos u}{\sqrt{1 - \sin^2u}}\ du$$

$$= \int_{-\pi/2}^{\pi/2} \dfrac{\cos u}{\sqrt{\cos^2u}}\ du$$

$$= \int_{-\pi/2}^{\pi/2} du = u\Big|_{-\pi/2}^{\pi/2}$$

$$= \dfrac{\pi}{2} - \left(-\dfrac{\pi}{2}\right) = \pi.$$

The particular beauty of this, is that just fore-armed with the knowledge that:

$$\int_{-1}^{1}\ ds$$ (the integral above, in terms of its arc-length $s(t)$) is indeed SOME real number, we can DEFINE:

$$ \pi = \int_{-1}^{1}\dfrac{dt}{\sqrt{1 - t}}$$

One can go on to define the usual trigonometric functions in terms of the invertible function (one thinks of this function representing "angle"):

$$A(x) = \int_{-1}^x \dfrac{dt}{\sqrt{1 - t}}$$ (defined on $[-1,1]$).

putting trigonometry on a rigorous analytic basis (although it does take a bit of work to recover some of their usual properties).