I’ve spent a lot of time trying to prove things.

With diagrams and algebra, back and forth between clipboard whiteboard blackboard & keyboard.

I can’t talk about what it’s like for a good mathematician but I can talk about it for a hack.

I can prove true & interesting things occasionally but only after wallowing in it for a long time, and after a half-dozen proofs of things that I later realize are not interesting or not true.

The representation of a problem becomes gradually more abstract.

I start with a full set of equations, then I gradually omit things that seem inessential, and I introduce a new symbol which feels like it denotes something but I’m not sure what, and then another higher level of abstraction, a jenga tower to the ceiling. I’m holding it very delicately and it’s crucial not to lose that concentration as I can see it teeter. Sometimes I stare out the window too long and when I look back down at the page the symbols no longer make sense: was this a matrix and that a vector? Why was I multiplying them together in this weird way?

I take a problem and present it to my intuition to see if there’s a glimmer of recognition.

I show it from different angles: vary the notation, write worked examples, draw arrow diagrams, draw venn diagrams, hoping that one of these approaches will trigger a memory of something similar.

I walk around the house checking the windows and doors. I check whether the answers are in the right units: dollars/person, quantity/year, and I check whether the signs go in the right direction.

I lose track of which way the inequality should be pointing, left or right.

My intuition doesn’t give me a strong signal so I just choose one and stick with it hoping that once I arrive at the conclusion I can work backwards and fix the directions.

Margaret Bray told me that, when she worked correcting Stiglitz’s proofs in the 1970s, she would sometimes find an odd number of sign errors instead of an even number. When that happened she would tell him it and he would quickly rewrite the discussion of that result: substituting an intuitive explanation of why the effect of \(\theta\) on F was negative for what had been an explanation of why the effect of \(\theta\) on F was positive.

Sometimes the goal is to maximize the distance between the assumptions and the conclusion.

I’m trying to make it seem more impressive. At first the starting point and end point are within sight of each other, and I hack away at either side, loosening the assumptions, tightening the conclusions, until the path stretches as far as possible.

I sometimes end up chasing what looks impressive. I twist the wording of the assumptions to make them seem weaker than they are, the conclusions to seem stronger, or hide ancillary assumptions in the body of the derivation. Looking back to some of my old papers I see signs of this and I’m ashamed of it.

There are a half-dozen different things I need to keep ambiguous.

I’m not sure which disambiguation will be the right one - whether to use strict or weak preferences; whether the numbering should start at zero or one; whether naturals or wholes or integers. Making a decision at one stage has implications for others. This part is Sudoku, finding something which fits many constraints.

I switch between two modes: is this true? How do I show everyone it’s true?

In the second mode it’s just trying to keep everything straight to nail it down but then occasionally a sickening feeling arises that it’s not my clumsiness that’s holding me back but perhaps the thing isn’t true after all, and I have to return to the first mode.

Sometimes I’m stumbling through equations trying to derive one from another but I have lost track of what they mean.

Like going through a room in the dark grasping for a door handle. I want to go back and draw some diagrams to get an intuition of what’s going on, some mental picture, but sometimes I try and try and nothing comes.

My coauthor suggests modifying an assumption in the setup.

Taking apart a Lego car to rearrange the unsatisfactory front wheels but it’s hard to anticipate all the downstream consequences of that; I don’t remember all the considerations I had in mind when I put together the front wheels.

Eventually the relief of finding the hidden door.

Suddenly I see the inequalities holding down variables like saxophone keys, which gives me the lemma: a condition for a set of inequalities between the elements of two vectors to imply an inequality between the sums of the vectors. I don’t need to write it down, I can take a break and go for a run, it’s enough to remember the idea of saxophone keys. Later I find out this is called Hall’s marriage theorem.

I’ll be writing out an equation for the 3rd time and suddenly a dusty image will be released.

When I quiet my brain to concentrate on just one thing the background noise becomes audible. I can hear the mice scratching in the walls. Not just the familiar memories, unfamiliar long lost memories come out too.

A sparse park in Manchester, a man who blanked me in a corridor in London, a cheese shop in Stockholm.

My desk is covered with notes: should write up what I have or push on towards the summit?

It’s a dangerous tradeoff. In retrospect it seems like I’ve often made the wrong decision: too-often tried for the summit, failed, and then left without a proper writeup. I come back to make another attempt, see my old footsteps and regret that I didn’t spend more time hammering nails into the rock.

Sometimes a gap opens up between what’s important and what’s orthodox.

If I make an assumption that is somewhat stronger than typical then everything is much more elegant and it feels like I can keep focus on what’s important. But then I get memories of being in seminars and seeing the speaker making an unorthodox assumption and how the energy drains out of the room.

For a while I tried to solve things falling asleep.

Once, in Birkbeck in 2008, as my mind untethered itself from its dock and I suddenly glimpsed that the integral could be taken vertically instead of horizontally. The problem was solved. For a few months afterwards I would put my book aside before I was ready to go to sleep and bring to mind some other unsolved problem but it never worked as well again.

More memories are released.

Beer at the dreary Birkbeck bar: we were all grasping around for things to say to each other to fill up an evening with. When we heard the kids roaring by the pool table we all looked up.

Saturday morning on the way back from the vegetable market in Farringdon. I sit down with a sandwich in a churchyard and I see my German neighbors are there too. I’m self conscious that my loneliness is exposed.

Making Will’s grandmother a shandy of beer and lemonade. Noticing the shadows of the roses in her garden, how they would change when I stood near them, the proximity of my own shadow would somehow make the shadows of the roses come into focus.

Perhaps 1/3 of the time is actually concentrating on the problem.

If it’s writing or programming I can just bring up a window and type away. If it’s deriving things then my mind is constantly drifting, trying to find some other path to go down – in reverie, doodling Zeus, refilling my pen, or looking something up.

On a good day it’s like swimming in cold water.

I don’t want to get in but once I’m in I don’t want to get out.

Once my chair is firmly pushed in at the desk I don’t want to get up to get a snack, to check the mail, I don’t want to go to the bathroom, I’ll be working through proofs while shaking my legs trying to stretch my bladder out.

I procrastinate starting work on a revision but once the document is open and the seal is broken I see things to fix and it’s difficult to stop.

When daughter was a baby she would push away a bottle of warm milk until the nipple was in her lips and then she’d clutch it tight while she’d drink.

On a bad day I’m rotating slices of a Rubik’s cube.

I’m checking to see if it’s solved then rotating again in some other direction.

Often I’ll over-estimate what I can accomplish.

I’ll have some cocky confidence that, this time, I can prove something I’ve failed to prove before. Or I’ll have a sketch and decide that it’s good-enough, that I can fill in the details later. Then later get a sickening feeling discovering that it’s really nothing, the bits omitted from the sketch were exactly the difficult parts. The voice in my head that causes these over-optimistic judgments causes other griefs in my life I think.

I get flashes of recollection of other peoples’ seminars and papers.

They seemed so boring and now my paper seems so similar to theirs.

The ideal state of mind has both (1) the clarity of fresh eyes; and (2) the suppleness of familiarity.

For that reason it makes sense to spend an entire day thinking about it, not little blocks of hours.

Weaselly unworthy thoughts bubble up.

I find myself thinking that a reader will be impressed by the quantity and the complexity of the derivation and give me credit for that.

When I’m trying to concentrate on something my weasel thinks of something I could order on Amazon.

I can suppress that. My weasel tries to get me to look up how to do a LaTeX symbol that would be useful (delta over equals). I can suppress that. The weasel sees a pair of brackets that could have an extra space. I give in. I fix the brackets. But when I’m fixing brackets I’m more vulnerable to each of the other temptations, and then after a while the weasel tells me it’s almost lunchtime. I look back on my morning it looks like swiss cheese.

I’m a grown adult but my concentration is still not under my control.

I put out treats for it, entreating it like a dog. Teach it bad habits. Give in to its whims.

When I switched from programming to studying economics I missed those numb hours that would pass by.

Some people seem to get into that state when they’re deriving things but it’s difficult for me to achieve that.

When you’re programming you get incremental feedback: you can see the mountain peak and you’re slowly getting closer to it. With proofs you’re going through the jungle and you don’t know if you’re getting closer or farther away. You could be on the brink of emerging into a clearing or it could be months more in the forest.

Why am I doing this?

As I’m feeling around a problem I think of a guy I knew from seminars at Harvard who would pause when a new slide came up and then nod quietly when he understood. Why do I spend my time at this when so many other people are better at this than me? Am I doing this for intrinsic or extrinsic reasons? One instinct is to untangle the threads of means and ends to see where they lead, another instinct tells me to leave it tangled, and I trust the second instinct more.

I look up from the corner I’m stitching.

I see the quilt goes to the end of the bed, out the window, across the fields all the way to the horizon.

Reading through an old draft like a landlord returning to his estate after a long trip.

To be reminded of the state of things. I’m more often surprised unpleasantly than pleasantly. My memory must be pickling things in a sweet vinegar.

Coming back to writing a paper after 5 years at Facebook I had a different kind of confidence.

Old anxieties were about failing to do things in the orthodox way. Now I feel that if it makes sense to me I won’t be embarrassed by it. I might be wrong but I’m not a fraud.

I’m comparing myself, and it seems unhealthy, but then also necessary.

Am I clearer sighted than I was last month? My decision about whether to attempt to solve this problem depends on that judgment.

I see that the problem I’m attempting is one that was notably not solved in a paper by Benabou and Tirole: whether I should give up now depends on how I assess my ability compared to theirs.

I’ve spent two hours trying out different matrices to see if they have a certain property, looking for a generalization about which matrices satisfy it, I still haven’t got one and now I don’t recall why I felt this was important – how much do I trust my decision to go down this road, and at what point do I go back to reconsider whether this problem is worth solving?

In Tel Aviv I would swim in the sea in the mornings.

By the third week I was still going down to the beach, getting in the water, and starting to swim, but after 30 seconds I would let my legs fall to the sand and just walk through the water staring into space, thinking. After a while I would remember to start swimming again.

Some things I proved

It’s likely that some of these had been proved before, but they each seemed useful enough to prove in the time and place.

The interpretation of an experimental outcome will depend on other outcomes.: Your estimate of the treatment effect on one outcome will depend on the observed effect on the other outcome, and the weight will be proportional to the difference in covariances between the treatment effects and the noise (if everything is joint Normal).
If item values are Normally distributed then selections will look like ellipses.: If you’re ranking items by predictions, e.g. ranking by \(p(A)+\beta p(B)\), and the probabilities have a joint Gaussian distribution, then the overall tradeoff between total \(A\) and total \(B\) will be an ellipse, equal to an isovalue of the joint distribution of \(F(p(A),p(B))\).
Modularity in the brain will cause characteristic inconsistencies in decisions.: If information is dispersed in the brain, and aggregated sequentially, then (a) mistakes will occur when there are interactions in your posterior (nonseparabilities), and (b) certain characteristic inconsistencies which will reveal those nonseparabilities.
Implicit preferences can be inferred from choices. (with Jon de Quidt): Observing choices over outcomes which differ in attributes can reveal the difference between implicit and explicit preferences – where implicit preferences can be due to either (a) tacit knowledge, (b) signaling motives, or (c) constraints on which decisions are allowable.
Noisy signalling advantages senders. (with Ines Moreno de Barreda): Suppose you’ll admitting a student to your PhD program only if they have a GRE score sufficient to imply their ability is above median. If students can exert effort to inflate their scores then you’ll end up admitting more than half of the students, even if you rationally adjust for the inflation.

Anatomy of a Mistake

I missed a malignant ambiguity because there was another ambiguity next to it, and that other ambiguity was benign.

There were two versions of a claim, and I was a little vague about which I was assuming:

\[\theta(x,x') = \theta(x'-x)\tag{A weak}\]

\[\theta(x,x') = \theta(|x'-x|)\tag{A strong}\]

My conclusion required the strong version but my assumptions only justified the weak version.

Why did I miss this? Because there was another ambiguity that was floating around, claim B:

\[\theta(x,x') - \theta(x+1,x') > 0. \tag{B}\]

It happens that assuming \(\mu=0\) (never mind what that is) implies both (B) and (A-strong). My thought process:

I don’t want to restrict to \(\mu=0\).
I knew that (B) is not true for all \(\mu\), but it is true for sufficiently small values, i.e. \(|\mu|<k\) for some \(k\).
So I was juggling two versions of an assumption about \(\mu\) (\(\mu\) is zero, \(\mu\) is small), and two versions of (A).
When my mind wandered onto whether claim (A strong) was justified I was reassured by remembering that it’ll work when \(\mu=0\) (true for both claim A and B), and additionally that for \(\mu=\varepsilon\) it’ll be OK (true only of claim B).

The key thing: if claim B had not have interfered I wouldn’t have made this mistake.

In the same way a pickpocket will wait for you to move before taking something off you, because the friction you feel on your buttock is attributed to the walking, not to the wallet leaving your pocket.