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How to Read Proofs Faster: A Summary of Useful Advice

July 4th, 2014 · 17 comments


Math Problem

The Wisdom of  (Math Nerd) Crowds

A couple weeks ago, I complained that my academic paper reading speed was slower than I would like given its importance to my productivity. I asked for your advice and you responded with over 60 comments and numerous private e-mails.

My goal in this post is to synthesize the best ideas from this feedback, as well as the results of my own self-reflection, into a clear answer. In particular, I’ve identified three big ideas relevant to trying to read technical papers — and in particular those containing mathematical proofs — as efficiently as possible.

Idea #1: There are no magic bullets…

This conversation helped cement an idea that I’ve long suspected to be true (but sometimes resist):

To develop a detailed understanding of a published mathematical proof is an ambiguous process that requires multiple attacks and can take an unpredictable amount of time (not unlike proving something in the first place).

As a result, you must be selective about what proofs you decide to dig into, as the time commitment is potentially serious. In the study of algorithms (my field), for example, in most cases when reading a relevant paper it’s sufficient to dive down just deep enough to answer the following questions:

  1. What is the main result and how does it compare to what was known before (or what a naive approach would provide)?
  2. What is the high-level insight/trick deployed in the bound argument that enabled this improvement?

With experience, I’ve found that I can consistently produce this level of understanding within an hour (sometimes less if the paper is well-written or building on my own results).

This knowledge is not enough by itself to deploy or extend the technique presented in the paper, but it is enough to recognize future opportunities where this technique might be relevant to a problem you care about (at which point, you’ll have to dive deeper). In other words, maybe just one out of ten papers you read will end up proving directly useful to your own work, so it makes sense to learn just enough from the papers you read to identify whether or not they’re in that crucial 10%.

Idea #2: There are ways to be more efficient…

If you must understand the details of a proof, then in addition to the high-level suggestion from above of preparing yourself psychologically for a difficult battle, the following low-level strategies might also help:

  • Instead of trying to read through the proof linearly, build a hierarchy of dependencies among the lemmata and theorems. Summarize each lemma and theorem in your own words and summarize each dependency relation; e.g., how does this theorem use the following three lemmata? Once you have this map, it becomes clearer where to begin a deeper dive and provides context for what you’re reading.
    (Last time I deployed this full proof-mapping process — which can be quite arduous — I ended up uncovering a flaw in a reasonably well-cited paper.)
  • In general, you should never start reverse-engineering a mathematical derivation until you understand what it is trying to show, why you expect it to be true, and how it will be used. If possible to assign some of this reverse engineering to a grad student, do so: it’s helpful to both parties.
  • Create your own system of notation and rewrite the relevant statements and re-derive the main results (or, rough approximations of the main results) using this notation. You’ll likely have to revise this notation system many times before you’re done, but this process will make it much easier for you to conceptualize the deeper insights of the argument.
    (I had to do this last week for a proof that I needed to understand better. It took me something north of six hours to complete! But I do certainly understand better now what is going on underneath the covers of this particular line of thinking.)
  • Form reading groups with like-minded academics. Something about collaboration has a tendency to bust open mental road blocks and incite more creative thinking.

Idea #3: But perhaps the best strategy of all…

Get the authors on the phone or pull them aside at a conference and have them walk you through the argument. Nothing is more efficient than having the original author fill in the details of his or her thoughts.

(This latter strategy, of course, becomes more available as your status in your field grows. It might not be advisable, for example, for a first year PhD student to apply it with too much regularity!)

The Concrete Satisfaction of Deep Work

June 21st, 2014 · 19 comments

ShopClassBookDeep Work as Soulcraft

I recently reread Matthew Crawford’s 2009 book, Shop Class as Soulcraft. Though Crawford’s primary goal is to make a philosophical case for the skilled trades (think: Mike Rowe with footnotes), a lot of what he writes resonates with my thinking about deep work.

Consider the following quote, which caught my attention:

“The satisfactions of manifesting oneself concretely in the world through manual competence have been known to make a man quiet and easy. They seem to relieve him of the felt need to offer chattering interpretations of himself to vindicate his worth. He can simply point: the building stands, the car now runs, the lights are on. Boasting is what a boy does, because he has no real effect in the world.” (page 15 of hardcover edition)

Cannot the same thing be said about any deep effort that results in the production of something too good to be ignored?

The reason, I think, that deep effort holds an appeal is that so much in modern knowledge work reduces to Crawford’s chattering interpretations — responding quickly to e-mail threads, bullet point self-promotion in PowerPoint slides, relentless online branding and ceaseless networking.

At some point, we tire of the shallow – necessary as it might be – and foster a desire to retreat into depth, create the best possible thing we’re capable of creating, then step back, point, and remark simply: “I did that.”

My Deliberate Quest to Read Proofs Faster

June 16th, 2014 · 64 comments

Deconstructing Theory

As a self-observant theoretician, I’ve learned that my research success depends on two intertwined factors: (1) my ability to digest and understand diverse results in my field; and (2) my ability to persistently attack good problems once identified.

Through practice over the past few years, I’ve become adept at the second factor. My deep work hours per week are quite high and have recently led to a correspondingly high rate of producing publishable results.

A nagging concern of mine, however, is that I’m not as good with the first factor. Indeed, I’m often frustrated with how long it takes me to digest interesting new results (and how often I end up aborting the process).

This concerns me because in my field voracious reading is required to keep the pipeline of good problems full.

What’s going wrong?

Read more »

Don’t Fight Distraction. Make It Irrelevant.

June 5th, 2014 · 30 comments


The War on Attention

My friend Dale (whose Ancient Wisdom Project blog you really should read) recently pointed me toward an interesting David Brooks column. In it, Brooks addresses the difficulty of maintaining focus in a distracted age:

And, like everyone else, I’ve nodded along with the prohibition sermons imploring me to limit my information diet. Stop multitasking! Turn off the devices at least once a week! And, like everyone else, these sermons have had no effect.

What’s interesting about this column is Brooks’ solution, which articulates a point that I firmly believe:

The lesson…then, is that if you want to win the war for attention, don’t try to say “no” to the trivial distractions you find on the information smorgasbord; try to say “yes” to the subject that arouses a terrifying longing, and let the terrifying longing crowd out everything else.

This rings true with my research on deep work. Those who are best at this skill are without exception obsessed with something that demands sustained attention, be it chess playing, writing, or theoretical physics. These deep workers rarely seem worried about distraction because it’s simply not an issue for them.

A New Focus on Focus

Distraction, from this perspective, is not the cause of problems in your work life, it’s a side effect. The real issue comes down to a question more important than whether or not you use Facebook too much: Are you striving to do something useful and do it so well that you cannot be ignored?

David Brooks would wager (and I would tend to agree) that once you can get to a positive answer to this question, you’ll find your worries about distraction rendered irrelevant.


I took the picture above in the woods near Georgetown where I like to go to churn on particularly knotty problems. As an interesting case study in the patience required for deep thinking, I originally posted the image back in January, where I talked about starting to work through an interesting but hard problem. Five months of persistent thought later, I finally finished the result. The deep life, it seems, is not a good fit for those who like immediate gratification!

Deep Habits: My Office in the Woods

May 28th, 2014 · 18 comments


Fashionably Deep

A reader sent me the above image. She implied that it represents a logical conclusion to my ever intensifying quest for depth.

I’m not there yet, but she’s not far off…

Case in point: I recently found a new hidden work location here on the Georgetown campus that I think trumps any previous spot I’ve found in terms of its ability to eliminate distraction and foster depth:

outside office 477px

I hate to give away all my secrets, but the location of this particular spot involves the Glover park trail that abuts the western edge of the medical school.


As an unrelated logistical note, my good friend Ramit Sethi is holding a webinar to explain what the hell goes on in that fabled Dream Job course he offers. If you’re interested, I believe it’s tonight (Wednesday, 5/28). You can learn more here.

Should We Work Like Novelists?

May 17th, 2014 · 46 comments


The Habits of a Writer

As I continue to clear out my queue after my end-of-semester blogging hiatus, there’s another article to mention that recently caught my attention. The fantasy author George R. R. Martin, it turns out, writes his books using Word Star, an ancient word processor that runs on DOS (see the screenshot above).

“It does everything I want a word processing program to do and it doesn’t do anything else,” he explained.

Martin is not the only fiction writer with idiosyncratic rituals surrounding his work. Neil Gaiman, for example, famously does much of writing long hand, and Stephen King is very particular about his desk.

This interests me because fiction writers are the epitome of deep workers (to make any progress, fiction writing requires your full concentration), and many of them, like Martin, Gaiman, and King, seem to rely on unusual but well-honed habits to get them into this mindset.

A natural question arises from this observation: Should those of us who work deeply in other fields follow their example?

This has been on my mind recently. In my pursuit to improve my ability to work deeply, I’ve paid a lot of attention to issues like scheduling (e.g., blocks versus lists) and tracking (e.g., milestones versus hour tallies). Like many knowledge workers, however, I’m  haphazard about the physical details that surround this work. I don’t have a special location or special tools I always use. I don’t have a head clearing ritual or hike to a hidden glen to tackle my knottiest problems.

But the more I hear about the habits of professional deep workers like novelists, the more I wonder if I should.

One of my goals this summer is to experiment more with the intensity piece of deep work (as I introduced in a recent blog post), and working with depth-inducing  habits and rituals of the type described above should be part of this experimenting. With this in mind, if you’ve found any such behavior useful in your own (deep) work, let me know about it in the comments.


Unrelated note: My friend Laura Shin, who writes from, and has been nice enough to feature me in some of her articles, just published an ebook, The Millennial Game Plan, which collects the best of her writing. She touches on a lot of issues we like to discuss here.

Why Didn’t Dan Kois Quit Facebook?

May 16th, 2014 · 29 comments

facebookThe Facebook Cleanse

Earlier today, one of the most read articles on Slate was titled “The Facebook Cleanse.” It was authored by Dan Kois, a veteran writer who reviews books for Slate and contributes regularly to the New York Times Magazine.

The article opens with the following:

“For years, I’d been frustrated that Facebook felt completely useless to me. The signal-to-noise ratio was way too low…my feed was overwhelmed by randos: publicists I’d met at parties years before, comedians with whom I’d shared stages in 2004, siblings of high school classmates, readers I’d friended or accepted friend requests from in hopes of Building My Brand.”

At this point in the article, I hoped (fruitlessly) that Kois would then reach the following eminently reasonable conclusion:

“Then I realized that I was a grown man and a serious writer and the fact that I was devoting any thought to this weird, juvenile ad network cooked up by a twenty-year old was ridiculous, so I of course quit.”

Alas: I know better.

Kois instead detailed an elaborate and time consuming routine in which he carefully culled, over many days, his “friend” list down to something that felt less useless.


Here’s the thing: I don’t find social media worthless, but I’m floored by its universality.

It’s not surprising, in other words, that lots of people like Facebook, but it is surprising that there are so few people who don’t.

Neil Postman warned about this in his insightful (but somewhat overlooked) 1992 book, Technopoly: The Surrender of Culture to Technology. The danger is not when a given technology tries to take over our culture, he warns, but instead when we stop debating the incursion and just assume it’s necessary.

(To be clear, I don’t know Kois, and suspect he probably has good reasons to use Facebook. I’m just using him an archetype to drive this discussion.)

Richard Feynman Didn’t Win a Nobel by Responding Promptly to E-mails

April 20th, 2014 · 18 comments

Feynman’s Faux Irresponsibility

Around 38 minutes into the above interview, the late physicist Richard Feynman describes an unorthodox strategy for defending deep work:

“To do real good physics work, you do need absolute solid lengths of time…it needs a lot of concentration…if you have a job administrating anything, you don’t have the time. So I have invented another myth for myself: that I’m irresponsible. I’m actively irresponsible. I tell everyone I don’t anything. If anyone asks me to be on a committee for admissions, ‘no,’ I tell them: I’m irresponsible.”

Feynman got away with this behavior because in research-oriented academia there’s a clear metric for judging merit: important publications. Feynman had a Nobel, so he didn’t have to be accessible.

There’s a lot that’s scary about having success and failure in your professional life reduce down to a small number of unambiguous metrics (this is something that academics share, improbably enough, with professional athletes).

But as Feynman’s example reminds us, there’s also something freeing about the clarity. If your professional value was objectively measured and clear, then you could more confidently sidestep actives that actively degrade your ability to do what you do well (think: constant connectivity, endless meetings, Power Point decks).

Put another way: if other knowledge work fields judged merit with the academic model, you’d probably find it a lot harder to get people to show up at your next project status meeting…even if you promised extra-fancy animations in your deck.


Hat Tip: Eric S.