The Missing Puzzle Piece

Recently, I decided to go back to basics, and learn mathematics from the ground up. You will find that post here. Calculus was first on the list. I’m taking the MIT course in single variable calculus, it’s available for free online.

It’s tempting to try and jump a few rungs. We understandably want to learn the fancy applications, the impressive stuff. Instead, I’ve chosen a more mundane route. Getting a strong handle on the basics. Understanding deeply the foundations is underrated. Josh waitzkin, a chess “prodigy” and martial artist, puts it brilliantly in his book:

“[We] began our study with a barren chessboard. We took on positions of reduced complexity and clear principles. Our first focus was king and pawn against king—just three pieces on the table… Layer by layer we built up my knowledge and my understanding of how to transform axioms into fuel for creative insight… This method of study gave me a feeling for the beautiful subtleties of each chess piece, because in relatively clear-cut positions I could focus on what was essential. I was also gradually internalizing a marvellous methodology of learning—the play between knowledge, intuition, and creativity.”  

Josh Waitzkin

Today I began with calculus, a subject that I’ve often touched on but never deeply understood. What is a derivative? Well (on the geometric interpretation) it’s the general formula for slope of the tangents to a curve. What is a tangent to a curve? Well if you take two points on a curve, and take the slope of that line, then push the points arbitrarily close together (take the limit as the difference between the points approaches 0), that’s a tangent line. I was beginning to get into the flow of the lecture. I already know what a derivative is, but I never had such a clear handle on why the formula is how it is. Then we started to solve the standard formula for taking the derivative of y=x^n. Although common place, I’ve never derived it myself. I’m feeling excited by now.

Suddenly, the flow stops. “…by the binomial theorem we have such and such”, the lecturer continues. I couldn’t remember the binomial formula. Worse, I didn’t understand it in this context. I didn’t know why the terms were exactly that way, and how the lecturer had found them. And a “whole load of junk” was a mystery to me. What was it? and Why was it junk?

I paused the lecture and jumped on to another website. Time to find the root of this problem. I watched some videos on the binomial theorem. I mostly understood, but not quite. There was a funny combinatoric thing, which was familiar previous study, but I couldn’t exactly remember it. I went back further, building up my understanding. First permutations, then combinatorics, then back to binomials. Then to pascals triangle and binomials, and linking all three.

When I switched back to the lecture, I understood.

The once formidable formula looked mundane. I knew what the pieces were, and why. I understood what the “junk” was, and how the junk would disappear in the final solution. When I got to the end, I quickly finished off my notes, replicating the proof from memory. My two hour journey into beginner calculus had begun.

When something doesn’t make sense, more often than not, we are just missing a piece of the puzzle. As someone who never took math seriously in high school, I’m missing a lot of the early puzzle pieces. It’s rewarding to find the missing pieces and bring them together.

So I’ve Decided to Learn Math

I might want to do an Economics PhD

My undergraduate degree was in economics, now I’m studying a master’s in philosophy. I absolutely love philosophy, and I’m excited by the opportunity to dive in deep. Yet there are reasons I’d consider further study in economics, rather than philosophy. Importantly, the job prospects for even the most excellent philosophy students, after a PhD, are frankly dismal. Dismal chances if you’re world class, less for anyone else. In contrast, Economics PhDs seem to have no trouble finding work in the private sector if their academic hopes are dashed. It may not be easier to become a professor, but at least the 3-7 years you’ve invested are useful when applying for other jobs. Second, there are areas of economics which I find deeply interesting. The strategic interactions of game theory are fun, while the combination of mathematical theory and psychology in behavioural economics is also especially interesting. I believe philosophy makes an excellent choice for an undergraduate program, especially when combined with a quantitative discipline. Clear writing and logical reasoning are excellent skills to develop. I am not at all knocking those who pursue a doctorate in philosophy, well aware of the grim career prospects, because they love the pursuit of knowledge and are happy to “waste” those years doing something deeply meaningful to them personally.

“An economics PhD is one of the most attractive graduate programs: if you get through, you have a high chance of landing a good research job in academia or policy – promising areas for social impact – and you have back-up options in the corporate sector since the skills you learn are in-demand (unlike many PhD programs). ” 80,000 Hours

Economics PhDs require a lot of math

Recently I applied for an economics related research position. One clear piece of feedback was that I was lacking the math background. Likewise, PhD admissions in economics require a lot of mathematics. So, if a PhD in economics is even a consideration, I will need to learn some math. Although I did an economics undergrad, it was not nearly as in depth as what is expected by doctoral programs. In economics it is mathematical signalling that reigns supreme.

Exactly what Math is required?

My understanding is that I should at least take a course each in Single Variable Calculus, Multi-variable Calculus, Linear Algebra, and Real Analysis (see this post from Yale, for example). These are the four I hope to take. Many people also recommend courses in Statistics and Differential Equations, and other proof based courses. These would be useful if I had time, but these are optional, and I have my own research to do. I would also prefer to go through the foundations as deeply as possible, rather than rushing through more.

University or self-study?

There are at least three important factors when considering for me when deciding how to learn this material.

  1. Will I do it?
  2. How can I learn the most?
  3. How will I signal my new math ability?

The first question is actually the most important. Perfect is the enemy of good enough. So, will I actually get up, do the study needed, and carry out the plan. Whatever method I choose I need to be compelled to do the work. University learning has the built-in fear of failure, which I’ve found compelling enough in the past. University learning also makes it easy to demonstrate my understanding to others. University also provides all of the course material for a large fee.

However, over the past few years I’ve learnt how to learn things pretty well on my own. My method is to consume explanations, then try exercises, then I make flashcards with any exercise I didn’t solve correctly. I also write myself Feynman style explanations for any concepts I forget or find difficult. I add these as flashcards as well. This may sound like I’m just memorising the material, but that’s not the case. First, I will not put anything in that I don’t already understand. Second, I did essentially the same thing to learn Spanish, and found that having an extremely solid grasp of particular words and patterns allowed me to express myself freely. Rather than giving a superficial memorisation of the subject, the memorisation helps develop clear understanding. Getting the course material for self study proved exceptionally easy for the first two courses. These courses are available on MIT Open Courseware, including video lectures.

The hardest part when considering learning on my own is demonstrating my understanding, the signalling, after all, was the original motivation. My plan is to complete the two calculus courses online. Later, I will take the last two courses at my university as a non-award student, gaining a letter grade for each of them. This, combined with my blog about my experience, should hopefully be enough to convince even the skeptics that I’ve got what it takes.

What’s next?

I’m starting out with the first two courses, single and multivariate calculus from MIT open course ware. The video lectures are downloading as I type this.

I originally considered purchasing a textbook. However, I found this excellent list of resources for single variable calculus. Especially, there are notes and practice problems available here which I think will be enough for at least single variable calculus.

I’ll make sure to update this blog with how it’s gone.