Proving stuff yourself is hard, and my initial attempts were catastrophic. I wrote one proof that began with a stronger statement than the conclusion I was attempting to draw, rendering the proof useless even in the unlikely case it was otherwise successful. It was not otherwise successful. Jumping through several dodgy implications and logical errors, it eventually arrived at a conclusion that was substantially weaker than what I was attempting to prove. The worst part? I couldn’t even spot my own mistakes, and I thought the proof was probably quite good.
This post is about my approach to real analysis. In many areas of theoretical research, proving things is of paramount importance. Sadly, I didn’t have the foresight to take a heavy math curriculum as an undergrad, and like many, my skills were lackluster at best. My solution? Prove every theorem in an intermediate analysis course.
Continue reading Trial by fire: My mission to prove every real analysis theorem from scratch
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:
Continue reading The Missing Puzzle Piece
“[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
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
Continue reading So I’ve Decided to Learn Math