To T or not to T?

I know, I know, I’ve been lazy recently posting nothing but my own YouTube videos. I promise I’ll write an updated report on my Successes At Doing Things quite soon.

Robin Does Everything – Back From the Psych Ward

I did a bit of remodeling on my YouTube channel. The Crazy Addict is no more – although I didn’t know it yet when I made this video – and in its place is a new series, which I will use primarily as a vlog to document my victories and defeats in my quest to relearn what my capabilities are after eight years of mental disability, and return to the arena of functional, productive human beings.

Algebra: On Why It Is Hard

Okay, I said I’d be writing only about analysis. But let’s have a little background on why you won’t find any algebra here, shall we?

Throughout school, math has always been simultaneously my favorite subject and my worst subject. I loved math, but math decidedly did not love me. This was primarily due to confusion over the mechanics of binary operations over different sets (, etc.). I could not for the life of me deeply understand the multiplication or addition of matrices or fractions, and only even began to understand subtraction of integers (for Pete’s sake!) once I’d taken calculus and learned to think of subtractions as distances on the real line. (Subtraction of non-integers never gave me any problems. Go figure.) During high school, I voraciously interrogated my older friends about their college math classes, and they, of course, told me only of what they thought I’d find “interesting” and “sufficiently college-math-y” — i.e., abstract and linear algebra, with some algebraic topology. I believed, then, that algebra was the entirety of Real Math (i.e. math-after-calculus). The view of Higher Math afforded by the math competitions I attended, Olympiad problems I looked up, and math camps I (unsuccessfully) applied to only reinforced this misconception. I dutifully bought a few abstract algebra textbooks, and enrolled in an online university linear algebra course during my senior year of high school, which I failed spectacularly.  Resigned to being irreparably Too Bad At Math to become a mathematician, I decided to study biopsychology in college, with an eye towards a career in neuroscience. I double majored in math because, masochistically, I was too in love to give it up just yet.

Then, the first semester of my freshman year, I got inexplicably, improbably lucky. What I thought was a routine item of homework given to me by my adviser turned out to be an open problem, and I had solved it. I had actually settled a much more general case than the stated problem, which I later discovered that convex geometers had considered too intractable to even mention. I was starting research in a neuroscience lab at the time as well – the math and psych departments were conducting a holy war for the rights to my seemingly-boundless-though-wildly-inefficient enthusiasm for research – but getting a major result in math convinced me to at least try to pursue a career as a mathematician. By the end of the year, I had dropped the biopsych major to focus entirely on math.

Doing this, as predicted, turned out to be absolute torture. In the spring of my freshman year, I took Math 52, the University of Vermont’s intro discrete course. Math 52 and I got along so poorly that within a couple weeks I dropped the course, and in order to gain the technical right to do so, switched my major from Pure to Applied. This meant that I needed to get a special override whenever I wanted to take an analysis course. The following semester, I took Math 251, Algebra I (Groups), and I honestly do not know how I am still alive to tell the tale. I barely scraped a C-. I never took Algebra II (Rings and Fields). I still couldn’t define an “ideal” if called to.

There are a few reasons, I think, why I find algebra so difficult. In no particular order:

1) Algebra is very jargon-intensive. And not intuitive jargon that you can picture, like “continuous” or “complete” or “converges.” No, jargon like “proper,” “regular,” normal,” “principal,” even “algebra” as a (pluralizable!) noun — words that have completely nonequivalent definitions depending on the specific class of mathematical objects they are applied to. When my colleagues attempt to discuss algebra with me (because I am in the masochistic habit of asking, “What did you learn in your classes today?” “Ooooh, you just finished your homework? Tell me about it!”), I have to stop and ask the definition of every word that isn’t “assume”, “prove”, “that”, “is”, “a”, or “of”. I must then ask the definition of most of the terms contained in the definition. A two-sentence problem expands into several pages of definitions. And this is precisely how I was taught to solve undergraduate algebra problems: “Unpack” each word in the problem; write down each definition; if the solution doesn’t become obvious once you’ve done this, you’re stupid.

2) Algebra is highly nonvisual. I adore Joseph Gallian’s undergrad abstract algebra textbook; I hate Dummit & Foote. Why? Gallian motivates the introduction of each new structure with a real-world application, and Gallian contains pictures. I am a very, very visual learner. If I can’t see it, touch it, picture myself physically manipulating it, then I’m not going to understand it. And abstract algebra is, well… abstract.

Once upon a time, I was visiting friends at the University of Chicago, and sitting in their math lounge with two UChicago math majors I’d just met. They were discussing their research, which was in algebraic… something. I was eating a bag of Skittles and being frustrated at my inability to understand what they were talking about. Finally, I pleaded, “show me that in Skittles!” and emptied my bag onto the table. I was hoping that maybe, if the group elements could be Skittles, they could be arranged in such a way that I could see on the table in front of me some of the subtleties of their interactions. One of the students swept all the Skittles but one to the side, picked up the singleton Skittle, and  emphatically put it back down in front of me. “Bam! Trivial subgroup!” he exclaimed. They laughed.

I can see functions. I can see continua. I can see myself tracing my objects of study in a sandbox, and can deeply understand that if you try to count the grains of sand, or study the relationship between one particular grain and its neighbor, you’re doing it wrong. Lines and curves and measures and derivatives and integrals in the sand are necessary and natural. And indeed, we all live in a (disputably four-dimensional) sandbox. What meteorologist has time to count individual air molecules? What student draws a line on paper by consciously stippling an uncountable number of points? Can you imagine living in a world where time progressed in any way that could not be described as a flow? (What’s that? You say you can’t? Oh good, I have company.)

A piece of wood, a piece of brick, a piece of metal – any hunk of material you could hold in your hands in also continuous. Sure, there’s one of it, but you don’t count its length, width, height, or mass, do you? Sure, it’s bounded by two-dimensional surfaces, but just like a bounded interval on the real line, it’s still continuous. Cut it into pieces, and you may have given yourself something to count, but each piece is still continuous.

3) “Proof by obvious.” At least in undergrad algebra, most of the statements you’ll be asked to prove are so obvious that you’ll have no idea how they’re not just tautologies. Right now, I am doggedly working my way through UIC’s undergrad algebra textbook, trying once more to teach myself the material it contains, and I am wrestling with problems such as: “If and are functions, prove that if and are injections, then is an injection.” As a former high school debate team member, I ask: How do you argue something if you can’t even see any points your opponent might use to support his point of view?

If you resonate with this blog post, know that you’re not alone, you’re not bad at math, and you can still become a mathematician! If you can think of any other reasons why algebra is hard, please leave them for me in comments.