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.