Scottish Nous

Mathematical Beauty

Today I give you the loveliest equation in the world:

e^(i * pi) + 1 = 0

This is Euler's Identity.  (And yes, it's true.)  Feynman remarked that it was "the most remarkable formula in the world", but "loveliest" better expresses my feelings. 

I was shooting pool last week when I found out that my opponent was a mathematics graduate student.  I had attempted to puzzle out the identity earlier that day, but using natural logarithms to knock the exponents out of the sky wasn't working too well (and I surmised that it wouldn't work very well either, due to that pesky little imaginary number).  Fortunately, it turned out that I was justified in giving up so quickly.  I asked this kid what was going on, and he instantly spit back the answer:

e^(i * x) = cos(x) + i * sin(x)

It's nice to know that the mathematics graduate students are learning things here at CU, because I'm really not too sure about some of the folks in the humanities.  If you're mathematically inclined, I have also blogged on some properties of the complex plane way back here.

Full-On Geek Induction

If you don't know about the badly behaved Polly Nomial, you haven't yet been inducted:

Once upon a time (1/t) pretty little Polly Nomial was strolling across a field of vectors when she came to the boundary of a singularly large matrix. Now Polly was convergent, and her mother had made it an absolute condition that she must never, ever enter such an array without her brackets on. Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the basis that it was insufficient, and made her way in amongst the complex elements.

Finish the truly horrifying tale for a mathematical moral.  Disclaimer:  This is manifestly not, and in the strongest possible sense, pure mathematics.

Philosophy for Truth's Sake?

According to legend, Thales demonstrated that the visionary philosopher didn't turn to the pursuit of truth for its own sake simply because he was incapable of functioning like any other otherwise successful member of society.  He allegedly demonstrated this by investing in olive (or wine) presses before a boom season on the basis of his mathematical predictions regarding the weather and harvest quality for that year.  I like to think that Thales then donated that money to charity and returned to an impoverished life devoted solely to astronomical, philosophical, and mathematical pursuits.

Now instead of demonstrating the same thing myself, I'm going to tell you how you can get rich quickly by following Thales lead.  According to a vast number of irrefutable recent studies, statistical prediction rules (SPRs) consistently outperform so-called experts in tasks as esoteric as determining the future quality of red Bordeaux wine long before it has finished aging.  In other words, it's about time for Robert Parker to step aside.  We have conclusive scientific proof that an algorithm renders Parker's refined palate obsolete.  (See Trout and Bishop's .pdf article here.)

As it turns out, most folks, particularly the experts who are losing out, find this difficult to believe.  So here's how you strike it rich.  You develop an SPR and bet against mere mortals.  Ideally, you go about this by seeking out some venture capital and founding a company with a futuristic yet classy name.  Then you employ several mathematicians, computer scientists, statisticians and bookies.  The bookies know who's betting the most money and on what.  (Or you could just start with football.)  The mathematicians, computer scientists, and statisticians produce your SPR.  Now you're done.  Bet away and watch the money roll in.   Once you've made enough money, you start replacing so-called experts with SPRs.  Just think how many companies would pay you for an algorithm that was proven to reduce error rates, cut labor costs, save valuable time, diminish the number of lawsuits filed against them, etc...

I'm probably not the first person to suggest this investment, but neither was Thales the first person to invest in olive presses.  If all this is too much work for you, just buy stock in some such company.  I'm going to.  Oh, and if you strike it rich thanks to this blog, how about sending me a kickback?  I accept U.S., Canadian, and Australian Dollars; Euros; Pounds Sterling; and Yen  via the PayPal link in the upper left corner.  I will also accept an awesome blog design. =)

Disjoint Ramblings and Sets

Debate with my father over the existence of infinities (yes, plural) has prompted this interesting quote by Georg Cantor:

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

I don't like the bit about creation and sustenance, though I confess a suspicion that the infinite extent of space and time has something to do with the probability that we're here at all.  It occurs to me that my father might approve of this quote if he decided to fix the reference of "infinity" a bit differently than I take it Cantor intended.    A Cantor quote with even more religious overtones, though hardly one that could be construed as illuminating that paradox contradiction known as the Trinity, follows:

A set is a Many that allows itself to be thought of as a One.

Not to disparage Cantor, but we might also point out that he believed Sir Francis Bacon authored the alleged works of William Shakespeare.  Chris Shields was thinking about teaching a course on infinity here at CU until Oxford made him an offer he didn't refuse.  I think I might go take a look at what the math department is offering, since, aside from auditing a course in Decision Theory, I won't have much besides my thesis to work on.  Not to minimize the work I'm going to have to put into my thesis.

Not So Imaginary Numbers

The interesting zeros of Riemann's zeta function fall into one of two classes, or so Riemann famously intuited and then hypothesized.  Picture your standard graph with two axes, x and y.  Now let the x-axis represent the real numbers and the y-axis the imaginary numbers.  This is the complex plane.   

Now if you're a philosopher - particularly with nominalistic predilections - you might be afraid of normal numbers, not to mention imaginary numbers.  (Imaginary numbers are just normal numbers multiplied by i, the perplexing square root of -1.)  It was originally thought that i was just a useful tool for getting the math done, so long as it was discharged before the calculation was finished.  Yet matters appear otherwise.  Perhaps the following will serve to mitigate your concern.