Hammers and Nails

Link post

If all you have is a ham­mer, ev­ery­thing looks like a nail.

The most im­por­tant idea I’ve blogged about so far is Tak­ing Ideas Se­ri­ously, which is it­self a gen­er­al­iza­tion of Zvi’s More Dakka. This post is an elab­o­ra­tion of how to fully in­te­grate a new idea.

I draw a di­chotomy be­tween Ham­mers and Nails:

A Ham­mer is some­one who picks one strat­egy and uses it to solve as many prob­lems as pos­si­ble.

A Nail is some­one who picks one prob­lem and tries all the strate­gies un­til it gets solved.

Hu­man be­ings are gen­er­ally Nails, fix­at­ing on one spe­cific prob­lem at a time and throw­ing their en­tire toolkit at it. A Nail gets good at solv­ing im­por­tant prob­lems slowly and la­bo­ri­ously but can fail to rec­og­nize the power and gen­er­al­ity of his tools.

Some­times it’s bet­ter to be a Ham­mer. Great ad­vice is always a ham­mer: an or­ga­niz­ing prin­ci­ple that works across many do­mains. To get the most mileage out of a sin­gle ham­mer, don’t stop at us­ing it to tackle your cur­rent pet prob­lem. Use it ev­ery­where. Ideas don’t get worn down from use.

Re­gard­less of which you are at a given mo­ment, be sys­tem­atic be­cause Choices are Bad.

Only a Few Tricks

I am re­minded of a clas­sic speech of the math­e­mat­i­cian Gian-Carlo Rota. His fifth point is to be a Ham­mer (em­pha­sis mine):

A long time ago an older and well known num­ber the­o­rist made some dis­parag­ing re­marks about Paul Er­dos’ work. You ad­mire con­tri­bu­tions to math­e­mat­ics as much as I do, and I felt an­noyed when the older math­e­mat­i­cian flatly and defini­tively stated that all of Er­dos’ work could be re­duced to a few tricks which Er­dos re­peat­edly re­lied on in his proofs. What the num­ber the­o­rist did not re­al­ize is that other math­e­mat­i­ci­ans, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The sec­ond vol­ume of Hilbert’s col­lected pa­pers con­tains Hilbert’s pa­pers in in­var­i­ant the­ory. I have made a point of read­ing some of these pa­pers with care. It is sad to note that some of Hilbert’s beau­tiful re­sults have been com­pletely for­got­ten. But on read­ing the proofs of Hilbert’s strik­ing and deep the­o­rems in in­var­i­ant the­ory, it was sur­pris­ing to ver­ify that Hilbert’s proofs re­lied on the same few tricks. Even Hilbert had only a few tricks!

The great­est math­e­mat­i­ci­ans of all time cre­ated vast swathes of their work by ap­ply­ing a sin­gle pre­cious tech­nique to ev­ery prob­lem they could find. My fa­vorite book of math­e­mat­ics is The Prob­a­bil­is­tic Method, by Alon and Spencer. It never ceases to amaze me that this same method ap­plies to:

  1. (The Erdős–Kac The­o­rem) The num­ber of dis­tinct prime fac­tors of a ran­dom in­te­ger be­tween and be­haves like a nor­mal dis­tri­bu­tion with mean and var­i­ance .

  2. (Heilbronn’s Tri­an­gle Prob­lem) What is the max­i­mum for which there ex­ist points in the unit square, no three of which form a tri­an­gle with area less than ?

  3. (The Erdős-Rényi Phase Tran­si­tion) A typ­i­cal ran­dom graph where each edge ex­ists with prob­a­bil­ity has con­nected com­po­nents of size . A typ­i­cal ran­dom graph where each edge ex­ists with prob­a­bil­ity has a gi­ant com­po­nent of size lin­ear in .

It’s amus­ing to note that in the same speech, Rota ex­pounded the benefits of be­ing a Nail just two points later:

Richard Feyn­man was fond of giv­ing the fol­low­ing ad­vice on how to be a ge­nius. You have to keep a dozen of your fa­vorite prob­lems con­stantly pre­sent in your mind, al­though by and large they will lay in a dor­mant state. Every time you hear or read a new trick or a new re­sult, test it against each of your twelve prob­lems to see whether it helps. Every once in a while there will be a hit, and peo­ple will say: “How did he do it? He must be a ge­nius!”

Both mind­sets are vi­tal.

To be a Nail is to study a sin­gle prob­lem from ev­ery an­gle. It is of­ten the case that each tech­nique sheds light on only one side of the prob­lem, and by cir­cum­am­bu­lat­ing it via the ap­pli­ca­tion of many ham­mers at once, one cor­ners the prob­lem in a deep way. This re­mains true well past a prob­lem’s re­s­olu­tion—in­sight can con­tinue to be drawn from it as other meth­ods are ap­plied and more satis­fy­ing proofs at­tained.

Usu­ally even the failure of cer­tain tech­niques sheds light on shape of the difficulty. One clas­sic ex­am­ple of an en­light­en­ing failure is the con­sis­tent over­count­ing (by ex­actly a fac­tor of two!) of primes by sieve meth­ods. This failure is so se­ri­ous and un­fix­able that it has its own name: the Par­ity Prob­lem.

Dually, to be a Ham­mer is to study a sin­gle tech­nique from ev­ery an­gle. In the case of the prob­a­bil­is­tic method, a breadth of cheap ap­pli­ca­tions were found im­me­di­ately by sim­ply sys­tem­at­i­cally study­ing uniform ran­dom con­struc­tions. How­ever, par­tic­u­larly adept Ham­mers like Erdős up­graded the ba­sic method into a su­per­weapon by stead­fastly ap­ply­ing it to harder and harder prob­lems. Vari­a­tions of the Prob­a­bil­is­tic Method like the Lovász Lo­cal Lemma, Shearer’s en­tropy lemma, and the Azuma-Hoeffd­ing in­equal­ity are now canon due to the per­sis­tence of Ham­mers.

Be Systematic

The up­shot is not that Ham­mers are bet­ter than Nails. Rather, there is a place for both Ham­mers and Nails, and in par­tic­u­lar both mind­sets are far su­pe­rior to the wishy-washy blind me­an­der­ing that char­ac­ter­izes over­whelmed novices. There may be an end­less sup­ply of ad­vice—even great ad­vice—on the in­ter­net, and yet any given per­son should or­ga­nize their life around sys­tem­at­i­cally ap­ply­ing a few tricks or solv­ing a few prob­lems.

Tak­ing an idea se­ri­ously is difficult and ex­pen­sive. You’ll have to tear down com­pet­ing men­tal real es­tate and build a whole new palace for it. You’ll have to field test it all over the place with­out get­ting su­per­sti­tious. You’ll have to gen­tly titrate for the amount you need un­til you have enough Dakka.

There­fore, be a Ham­mer and make that idea pay rent. Hell, you’re the pres­i­dent, the em­peror, the king. There’s no rent con­trol in your head! Get that idea for all its got.

Ex­er­cise for the reader: all things have their ac­cus­tomed uses. Give me ten un­ac­cus­tomed uses of your fa­vorite in­stru­men­tal ra­tio­nal­ity tech­nique! (Bonus points for demon­strat­ing in­tent to kill.)

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