Skeptophilia (skep-to-fil-i-a) (n.) - the love of logical thought, skepticism, and thinking critically. Being an exploration of the applications of skeptical thinking to the world at large, with periodic excursions into linguistics, music, politics, cryptozoology, and why people keep seeing the face of Jesus on grilled cheese sandwiches.

Wednesday, November 15, 2017

Advanced elegance

I think it's a natural human tendency to be awed by what we don't understand.

I know when I see some abstruse concept that is far beyond my grasp, I'm impressed not only by how complex the universe can be, but that there are people who can comprehend it.  I first ran into this in a big way when I was in college, and took a class called Classical Mechanics.  The topic was why and how objects move, how that motion affects other objects, and so on.

It was the first time in my life I had ever collided with something that regardless of my effort, I couldn't get.  The professor, Dr. Spross, was a very patient man, but his patience was up against a classical-mechanics-proof brain.  On the first exam, I scored a 19.


And I'm convinced that he had dredged up the 19 points from somewhere so I wouldn't end up with a single-digit score. I ended that class with a C-, which I think Dr. Spross gave me simply because he didn't want me back again the following semester, spending another four months ramming my poor physics-deficient head up against a metaphorical brick wall.

There's one memory that stands out from that experience, nearly forty years ago, besides the overwhelming frustration.  It was when Dr. Spross introduced the concept of the "Hamiltonian function," a mathematical framework for analyzing motion.  He seemed so excited about it.  It was, he said, an incredibly elegant way to consider velocity, acceleration, force, momentum, and so on.  So I thought, "Cool!  That sounds pretty interesting."

Following that cheerful thought was an hour and a half of thinking, "I have no fucking idea what any of this means."  It was completely opaque.  The worst part was that a number of my classmates were nodding their heads, writing stuff down, and seemed to get it with no problem.

So I was either the only dumb one in the class, or they were just better at hiding their dismay than I was.

Anyhow, I think that was the moment I realized a career in research physics was not in the cards for me.

To this day, the "Hamiltonian function" remains something that in my mind symbolizes the Unknowable.  I have deep and abiding admiration for people for whom that concept makes sense (first and foremost, William Rowan Hamilton, who developed it).  And I'm sure it is elegant, just as Dr. Spross said.  But experiencing that elegance was (and probably still is) entirely beyond me.

It's this tendency to find what we can't understand awe-inspiring that has led to the idea of the god of the gaps -- in which gaps in our scientific knowledge are attributed to the incomprehensible hand of the divine.  Theologian Dietrich Bonhoeffer realized what the problem with this was, at least for people who are religious:
How wrong it is to use God as a stop-gap for the incompleteness of our knowledge.  If in fact the frontiers of knowledge are being pushed further and further back (and that is bound to be the case), then God is being pushed back with them, and is therefore continually in retreat.  We are to find God in what we know, not in what we don't know.
Anyhow, that was a long-winded preamble as an explanation of why all of this comes up in today's post.  I immediately thought of the awe-inspiring nature of what we don't understand when I read an article yesterday about two researchers at the University of Rochester, Tamar Friedmann and Carl Hagen, who found that a method for calculating the energy levels of a hydrogen atom generates the well-known number pi.

[image courtesy of the Wikimedia Commons]

It turns out to have something to do with a mathematical function called the Wallis product, which says that you can generate π/2 by a simple series of multiplications:
π/2 = (2/1) x (2/3) x (4/3) x (4/5) x (6/5) x (6/7) x (8/7) x (8/9)....
The pattern is that the numerators of the fractions are 2, 2, 4, 4, 6, 6, 8, 8... and the denominators 1, 3, 3, 5, 5, 7, 7, 9, 9...  And the cool thing is, the more terms you add, the closer you get to π/2.

Now, as for why this is so... well, I tried reading the explanation, and my eyes started spinning.  And I've taken lots of math courses, including calculus and differential equations, and like I said earlier, I majored in physics (as much of a mistake as that turned out to be).  But when I took a look at the paper about the energy levels of hydrogen and the Wallis product and gamma functions, I almost could hear Dr. Spross's voice, explaining it in a tone that implies that it would be immediately clear to a small child, or even an unusually intelligent dog.

And all of those feelings from Classical Mechanics came bubbling up to the surface.

So I'm left with being a little in awe about it all.  Somehow, even though I have no real understanding of why, the same number that I learned about in geometry class as the ratio between a circle's circumference and its diameter shows up in the energy levels of hydrogen atoms.  Predictably, I'm not inclined to attribute such correspondences to the hand of the divine, but I do think they're (in Dr. Spross's words) "elegant."  And even if I never get much beyond that, I can still appreciate the minds of the people who can.


  1. Further proof that you and I have separated in space-time - at birth in space and by more than a decade in time ;-)

    I, too, studied physics. Applied physics to be exact, and was set about to major in astrophysics until a nice paying job came my way and I realized that I was never going to go very far with my degree.

    You see, I was a C student at best. I mean, when I graduated with a non-specialized Bachelor of Science I was something like 0.1% above the minimum required overall average to avoid getting kicked out of school.

    Here's where we differ in just about the same way. While classical mechanics was one of my best classes (a solid B!), Electricity and Magnetism was hands down my worst.

    At our school if you didn't write a final exam instead of giving you a zero and calculating your final grade based on that - the final was always worth 50% of your grade or more so it was impossible to pass the course if you didn't write it - they would give you a 32%. This served two purposes: 1) it would ensure the registrar knew you didn't write the final, and 2) totally fucked up your average (or GPA in more American terms).

    Well, I wrote the final, which at the time was worth 70% of my final grade (10% lab work and 20% class assignments), and my final grade in the class was.... 30. Thirty percent. Three zero. I would have received a grade 2% better had I stayed in my apartment and slept through the exam than I did for dragging my ass out of bed at 8 A.M. and writing it.

    Nevertheless, I was determined to continue with my physics education. I persevered and took the class again. I went to every lecture (Tuesday's at 8 A.M. for three hours), did every assignment on my own, studied extra every week, and when the midterm exam came around I still failed it.

    That was okay though, because now I knew where to focus. So I doubled down on my efforts and prepared myself for the final, which due to my failure of the midterm was worth 70% of my grade - again.

    I had done okay on my labs. Averaging about 7/10 on them. My assignments acceptable sitting at about a 13/20. I needed just 31/70 on the final to be done with this class forever.

    I got 22/70. Nine marks short.

    Back then, if you failed a course twice they kicked you out of your program and I was not ready to give up my dream of being a physicist so into the professor's office I marched. I sat down in front of his desk and...


    I showed him my study notes and all the work I did in the margins for my assignments before I handed them in. I walked him through moments from every lecture so he knew I had been to them all, and when I was done he got my exam out and he went through it line-by-line asking me questions and scribbling stuff down on an engineering notepad beside him.

    When he was done he sat his pen down and he pulled out his final marks sheet (still on paper back then) and scratched out my failing grade and wrote "50" (the only way to get a 50 back then was to be given a fifty, so the registrar would know you were handed the credit. Even if you *actually* got 50 they would round it up to a 51).

    Then, he looked up at me and said, "Mr. Butters, if I sign this marks sheet as it is will you promise me you'll never take another one of my classes?"

    I looked him right in the eye and said, "As God as my witness, I promise."

    He smirked and replied, "He wasn't much help to you in my class so I'm going to need a better witness."

    Then he signed the marks sheet and waved me out of his office.