Trial by fire: My mission to prove every real analysis theorem from scratch

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.

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Two Approaches to Learning

Josh Kaufman argues you can learn anything in less than 20 Hours. This seems to contradict research that says it takes (on average) 10,000 hours to master a skill. Josh Kaufman doesn’t mean it takes you 20 hours to master a skill, it just takes that long to get “good enough”. This article explains two approaches and how to apply them effectively.

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Don’t Learn Latin, Don’t Use Translations, Don’t Be a Hack.

I recently read my first book in Spanish. To celebrate, I’m reflecting on how I got here. It was an arduous journey and there are many pitfalls to avoid. Three principles guided my way:

  • Don’t be led astray
  • Eliminate the unimportant
  • Go slow to go fast
Continue reading Don’t Learn Latin, Don’t Use Translations, Don’t Be a Hack.

Radical Democracy

How can governance represent the needs of the people?  How can governments disseminate power, and give equal consideration of interests? One proposal is quadratic voting, outlined in the book “Radical Markets”. The authors, Eric Posner and Glen Weyl, propose “Quadratic Voting”. This proposal replaces ‘one person one vote’. It replaces equal influence in choosing governments with equal influence on governance. Equal influence rather than equal say.

Under quadratic voting, everyone receives an equal number of voting credits. These credits can buy influence. Raising an issue for consideration or voting for it once raised. Or voting against an awful proposal, as the case may be. We would all have the same number of voting credits. We could all use them how we like. Perhaps climate change moves you. Or marriage equality. Or mental health policy. Or any number of things. You could your influence on what is most important to you.

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Fantastic Beasts and Where they Came From

Sitting in the botanic gardens, hungover, watching the ducks skim across the water, waiting for the revival we knew only coffee could bring. Attempting a normal conversation. “Have you heard the theory that octopuses are aliens that arrived on a comet?”, my friend asked. I hadn’t.

We examined this strange theory. It apparently relied on the absurdness of octopodes, fantastic creatures with distributed brains, unlikely intelligence, and the ability to squeeze and escape through a hole as small as their eye (see below). The theory was equally absurd, can this creature really be explained by accidental intergalactic space travel?

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