Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

Saturday, January 12, 2008


I often have to explain to people that I do math, not for it's scientific utility, but it's aesthetic value. Many people either get confused by this idea, or think of pretty math pictures or mystifying symbols. This is not at all what a mathematician is thinking of when he calls math "pretty".

A much closer representation would be Pope's question "Ah, Why, ye Gods, should two and two make four?" Why, indeed! There is a simplicity, a finality that is striking and counter-intuitive in math. It is obvious that 2+2=4, and saying otherwise will confuse people (even when you're correct.)

But mathematically obvious and intuitively obvious are two very different things. Mathematically obvious means "a proof is trivial". Intuitively obvious means "the assumption that it is true is so ingrained within us that another system is difficult to comprehend." 2+2=4 is not mathematically obvious... until you define addition, and define equality. 2+2=4 is so often used because of this fact. It relies, almost solely on the foundations of mathematics. If math were a building-- one built over millennia, with thousands of architects-- 2+2=4 would be the buttress opposite 1≠0.

There is something mystifying in this idea, something almost haunting. More haunting yet, is Euler's Identity. eiπ+1=0. This may be meaningless to my audience...
e (2.718281...) is the "natural base" It has a number of fascinating properties, perhaps the most important of which is that the the slope of the line
ex at any point is also ex. This number is primarily important in calculus (where it is the most important number): real analysis and the study of differential equations. π (3.14159) is the ratio of a circle's circumference to it's diameter. It is the most important number in trigonometry, and therefore all of geometry. Consequently, π, like e, shows up everywhere in math. i is the imaginary number √-1. Complex analysis is the study of this number.

We have, then, undoubtedly the 5 most important numbers in math: 0, 1, e, i, π. They come together in one simple equation-- 7th grade math-- eiπ+1=0. When you see this equation, you are simultaneously looking at all of analysis, as well as analytic geometry. If this doesn't leave you in some way uncomfortable, you have missed something: take a math class.

As you learn the required material-- certainly more work than 7th grade math-- the equation changes from meaningless symbols to utterly counter-intuitive. Mathematically, it's obvious; there are proofs which require no more than high school math; but intuitively, it cannot be. How can everything be related so simply?

If you still do not see it, if you still cannot feel it. Think of the way a good song can almost bring you to tears, or the way certain paintings refuse to be forgotten, or the power in a great piece of literature. That's what a mathematician feels when he sees this equation.

"Why must e to the i pi negate 1? And how, O God, does it steal my sleep?"

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