Mathematical Mindset

I agree that op­ti­miza­tion am­plifies things. I also agree that a math­e­mat­i­cal mind­set is im­por­tant for AI al­ign­ment. I don’t, how­ever, think that a “math­e­mat­i­cal mind­set” is the same as a “proof mind­set”. Rather, I think that the lat­ter is closer to be­ing a “pro­gram­ming mind­set”—or, in­deed, a “se­cu­rity mind­set”. And that a “math­e­mat­i­cal mind­set” is largely miss­ing from AI-al­ign­ment dis­course at pre­sent.

Whereas oth­ers see a di­vi­sion be­tween two clusters, of the form,

sci­ence/​physics

vs.

math­e­mat­ics/​pro­gram­ming/​logic

I, by con­trast, see a hi­er­ar­chi­cal pro­gres­sion that looks some­thing like:

sci­ence < pro­gram­ming < physics < math­e­mat­ics < logic <...

where, in this con­text, these words have mean­ings along the fol­low­ing lines:

sci­ence: things be­ing made of parts; decomposition

pro­gram­ming: things be­ing made of mov­ing parts; con­stant-ve­loc­ity mo­tion; causal networks

physics: things be­ing made of mov­ing spa­tial parts; ac­cel­er­ated mo­tion, ro­ta­tion, fluidity; substance

math­e­mat­ics: mod­els be­ing made of parts; tran­sub­stan­ti­a­tion; meta­physics; theorization

logic: con­cepts be­ing made of parts; time re­ver­sal; ontology

Of course I’m not us­ing these words stan­dardly here. One rea­son for this is that, in this dis­cus­sion, no one is: we’re talk­ing about mind­sets, not about so­ciolog­i­cal dis­ci­plines or even clusters of par­tic­u­lar ideas or “re­sults”.

But the re­ally im­por­tant rea­son I’m not fol­low­ing stan­dard us­age is be­cause I’m not try­ing to in­voke stan­dard con­cepts; in­stead, I’m try­ing to in­vent “the right” con­cepts. Con­se­quently, I can’t just use stan­dard lan­guage, be­cause stan­dard lan­guage im­plies a model of the world differ­ent from the one that I want to use.

It is com­monly be­lieved that if you want to in­tro­duce a new con­cept that is similar or re­lated (but—of course—non­iden­ti­cal) to an old con­cept, you shouldn’t use the same word for the new con­cept and the old, be­cause that would be “con­fus­ing”. I wish to ex­plic­itly dis­agree with this be­lief.

This view pre­sup­poses that ac­tively shift­ing be­tween mod­els of the world is not in our reper­tory of men­tal op­er­a­tions. But I speci­fi­cally want it to be!

In fact, I claim that this, and not proof, is what a “math­e­mat­i­cal mind­set” is re­ally about.

For math­e­mat­ics is not about proofs; it is about defi­ni­tions. The essence of great math­e­mat­ics is com­ing up with a pow­er­ful defi­ni­tion that re­sults in short proofs.

What makes a defi­ni­tion “pow­er­ful” is that it re­flects a con­cep­tual up­grade—as dis­tinct from mere con­cep­tual anal­y­sis. We’re not just try­ing to figure out what we mean; we’re try­ing to figure out what we should mean.

A math­e­mat­i­cal defi­ni­tion is what the an­swer to a philo­soph­i­cal prob­lem looks like. An ex­am­ple I par­tic­u­larly like is the defi­ni­tion of a topolog­i­cal space. I don’t know for a fact that this is what peo­ple “re­ally meant” when they pon­dered the na­ture of “space” dur­ing all the cen­turies be­fore Felix Haus­dorff came up with this defi­ni­tion; it doesn’t mat­ter, be­cause the power of this defi­ni­tion shows that it is what they should have meant.

(And for that rea­son, I’m com­fortable say­ing that it is what they “meant”—ac­knowl­edg­ing that this is a lin­guis­tic fic­tion, but us­ing it any­way.)

Notably, math­e­mat­i­cal defi­ni­tions are of­ten re­defi­ni­tions: they take a term already in use and define it in a new way. And, no­tably, the new defi­ni­tion of­ten bears scant re­sem­blance to the old, let alone any “in­tu­itive” mean­ing of the term—de­spite pre­sent­ing it­self as a suc­ces­sor. This is not a bug. It is what philo­soph­i­cal progress—the­o­ret­i­cal progress, progress in un­der­stand­ing—looks like. The re­la­tion­ship be­tween the new and the old defi­ni­tions is ex­plained not in the defi­ni­tions them­selves, but in the re­la­tion­ship be­tween the the­o­ries of which the defi­ni­tions are part.

Hav­ing a “math­e­mat­i­cal mind­set” means be­ing com­fortable with words be­ing re­defined. This is be­cause it means be­ing com­fortable with mod­els be­ing up­graded—in par­tic­u­lar, with mod­els be­ing re­lated and com­pared to each other: the ac­tivity of the­o­riza­tion.

It oc­curs to me, now that I think about it, that the term “the­o­riza­tion” is not very of­ten used in the AI-al­ign­ment and ra­tio­nal­ist com­mu­ni­ties, com­pared with what one might ex­pect given the de­gree of in­ter­est in episte­mol­ogy. My sus­pi­cion is that this re­flects an in­suffi­ciency of com­fort with the idea of mod­els (as op­posed to things) be­ing made of parts (in par­tic­u­lar, be­ing made of pa­ram­e­ters), such that they are re­lat­able to and trans­formable into each other.

This idea is, ap­prox­i­mately, what I am call­ing “math­e­mat­i­cal mind­set”. It stands in con­trast to what oth­ers are call­ing “math­e­mat­i­cal mind­set”, which has to do with proofs. The re­la­tion­ship, how­ever, is this: both of them re­flect an in­ter­est in un­der­stand­ing what is go­ing on, as op­posed to merely be­ing able to de­scribe what is go­ing on.