Set Up for Success: Insights from ‘Naïve Set Theory’

Foreword

This book has been re­viewed pretty thor­oughly already. Rather than restate each chap­ter, I’ll be shar­ing in­sights: some book-spe­cific, some gen­eral.

I am quite proud of my time-to-com­ple­tion for this book—just over a week, work­ing around a very stren­u­ous courseload. I went from hav­ing to fo­cus re­ally hard to pick up new con­cepts to read­ing no­ta­tion nearly as fluently as the English sur­round­ing it. The chap­ters started mak­ing sense—it felt less ran­dom, and more like a co­her­ent story wherein the pro­tag­o­nist slowly adds ow­ers to their ar­se­nal.

Naïve Set Theory

Functions

Func­tions are just static sets of or­dered pairs . They are not dy­namic in­dex­ing func­tions, they do not perform effi­cient lookup, please do not waste an hour of your life try­ing to figure out how you could do such a thing within the sim­ple ma­chin­ery af­forded by set the­ory up to that point.

This is one of those things that Nate talked about—how skip­ping over just one word a few chap­ters prior can cause you to waste hours. Dur­ing my con­fu­sion, I knew this was prob­a­bly the case, but I still couldn’t man­age to im­me­di­ately over­come my in­tu­itions of what a func­tion should be. This is one rea­son I’m open to work­ing through the MIRI Re­search Guide with oth­ers.

Families

Fam­i­lies are, iron­i­cally enough, just a spe­cial kind of func­tion; don’t let your in­tu­ition fool you—they aren’t “groups of func­tions”. A fam­ily be­long­ing to maps each el­e­ment of the in­dex set to an el­e­ment . For ex­am­ple, a fam­ily from to could be (thanks to Da­cyn for helping me clar­ify my writ­ing).

Zorn’s Lemma

I spent three hours star­ing at this proof. I un­der­stood what ZL meant. I grasped the rele­vant con­cepts. I read other ver­sions of the proof. I still spent three long hours on this damn proof, and then I went to my classes. I don’t know why I ended up figur­ing it out, but I sus­pect it was a com­bi­na­tion of two fac­tors: my brain worked through some things dur­ing the day, and I re­ally wanted it. On the bus home, I men­tally snapped and de­cided I was go­ing to un­der­stand the proof. And I did.

I’m pleased to share my de­tailed proof out­line of Zorn’s Lemma, the product of many hours of am­bi­ent ex­as­per­a­tion, rewrit­ten in my own no­ta­tion. Look­ing back, the proof in the book was pretty bad; it was nei­ther suc­cinct nor in­tu­itive, but in­stead im­posed a marais of men­tal vari­able track­ing on the reader. I think mine is at least a lit­tle bet­ter, if not fully fleshed-out at all junc­tures.

Proof Calibration

As some­one with­out a for­mal de­gree in math­e­mat­ics, it was im­por­tant for me to mon­i­tor how I ap­proached the ex­er­cises in the book. When­ever the au­thor be­gan a proof, I tried to gen­er­ate a men­tal proof sketch be­fore read­ing fur­ther. Some­times, I thought the proof would be easy and short, but it would turn out that my ap­proach wasn’t rigor­ous enough. This was valuable feed­back for cal­ibra­tion, and I in­tend to con­tinue this prac­tice. I’m still wor­ried that down the line and in the ab­sence of teach­ers, I may be­lieve that I’ve learnt the re­search guide with the nec­es­sary rigor, go to a MIRIx work­shop, and re­al­ize I hadn’t been hold­ing my­self to a suffi­ciently high stan­dard. Sugges­tions for ame­lio­rat­ing this would be wel­come.

Forwards

Anticipation

One fac­tor which helped me suc­ceed was that I en­sured my morn­ing read­ing was what I most looked for­ward to each day. I was ex­cited to go to sleep, wake up early, pre­pare a deli­cious break­fast, and curl up near the fire­place with book and pa­per handy. Triv­ial in­con­ve­niences can be fatal—do what­ever you must to en­sure you prop­erly re­spect and an­ti­ci­pate your study time.

Defense with the Dark Arts

The most use­ful pro­duc­tivity-re­lated ad­vice I ever read was by Nate Soares (Dark Arts warn­ing), and it re­lates to im­bu­ing your in­stru­men­tal goals with ter­mi­nal val­ues. Ever since hav­ing read that ad­vice, ev­ery te­dious as­sign­ment, ev­ery daily rou­tine, ev­ery keystroke—they’re all backed up by an in­tense de­sire to do some­thing about the pre­car­i­ous situ­a­tion in which hu­man­ity finds it­self.

In­ter­nal Light

If you don’t know where to start, I think the in­ter­nal fire has to be lit first—don’t try to force your­self to do this (or any­thing else) be­cause you should. Stop the guilt-based mo­ti­va­tion, proudly stake out what you want, and trans­form your life into a daz­zling as­sort­ment of ac­tivi­ties and tasks im­bued with your ter­mi­nal val­ues, your bright­est vi­sions for your­self and the fu­ture.