Here’s a roundabout analogy to try to convince you that the output of a reflection process you endorse may be different from your “true values.”
Suppose I give you a 2048-bit number N.[1] I ask you “What is the true prime factorization of N? Please use whatever reflection process you want to come up with the best factorization you can.”
You find out pretty quickly that N has the factors 2 and 7. You spend about a hundred years checking more factors according to your endorsed reflection process (running a factorization algorithm on the beefiest computer you can find), but you don’t find any other factors.
You come back to me and ask “is the true prime factorization 2 × 7 × A?”[2]
“No,” I tell you. “But here is some new information for you. Try dividing by B.”[3]
You check, and N is indeed divisible by B. “Yep, the factorization I guessed is definitely not the true prime factorization,” you say. “Is the true prime factorization 2 × 7 × B × C?”[4]
“Wrong again,” I tell you. “But I have even more information for you. It is written on this pocketwatch. Look closer at the pocketwatch as it swings back and forth in front of your face. It is making you so veeery sleeeeepy. That’s right. Now, when I snap my fingers, you will wake up believing that the true prime factorization of N is 2 × 2.”
I snap my fingers. “Oh, thanks for that information!” you say. “Now I know that the true prime factorization is 2 × 2.”
The point is, even the best reflection process you can think of may fail to account for some crucial information. And hopefully it can robustly tell the difference between helpful information and harmful information.
For example, maybe N = 29522110801023785555247567907018022843013371193486904872915694135366948906267412459560469419313468477571904190875078325307783298702278061314706021273052523914864561727670955407896206738948955813504747448172831328073078012451035444606017289679166070717156612947440897221609673043263408054415375773691379283198201987372931659507826484639961297915624514954455314101431489726823065604374788650066472170603904794973458618994986833512839575283873771252517988691292017425081313700740089351712559811486464802367263467854576668024443015614104081670018991747427099820025784949521876071608490248395046666723258743709356296535782.
A = 1232477336227426692380443210888895449621471079543761850047395308187049594617001861601696813319110513848274876635355987350103099368741162622794253649965690823778078933544136385470959159681614005099869579024125795246308841648113743911773571553129957340479245748931756917137130013760495971399023845580343786885491244375889710319394039047959654983736068473827495579018934560487915085510788809201773576234050572295508350789032949910040027280393874600876759921821682717244338109892239475602768230768732937311567202612318055134309177705320521639635449240702247038285504267829595346064686009110312205007737419845285089599211
B = 28404358936141244817111713617600331891793591357912402403090074101183255558265948779426267547155101307334436867856129404340927293405086432682892248263827662441988227069219008681660301942027627323842312874031058512179223840896157038194006799237133125684806392915413774710883917390425119272589884765032741213943
C = 43390429581540126017572442049413488959139702613459795717022804047186295570874599744101057201841640892276368639418940532012449598478611604251554159646145486573777182692131295315212900911195947383450995718772416566366046388412317400464871279622896378528747304275985094440653456156567729772192224995294814959277
Here’s a roundabout analogy to try to convince you that the output of a reflection process you endorse may be different from your “true values.”
Suppose I give you a 2048-bit number N.[1] I ask you “What is the true prime factorization of N? Please use whatever reflection process you want to come up with the best factorization you can.”
You find out pretty quickly that N has the factors 2 and 7. You spend about a hundred years checking more factors according to your endorsed reflection process (running a factorization algorithm on the beefiest computer you can find), but you don’t find any other factors.
You come back to me and ask “is the true prime factorization 2 × 7 × A?”[2]
“No,” I tell you. “But here is some new information for you. Try dividing by B.”[3]
You check, and N is indeed divisible by B. “Yep, the factorization I guessed is definitely not the true prime factorization,” you say. “Is the true prime factorization 2 × 7 × B × C?”[4]
“Wrong again,” I tell you. “But I have even more information for you. It is written on this pocketwatch. Look closer at the pocketwatch as it swings back and forth in front of your face. It is making you so veeery sleeeeepy. That’s right. Now, when I snap my fingers, you will wake up believing that the true prime factorization of N is 2 × 2.”
I snap my fingers. “Oh, thanks for that information!” you say. “Now I know that the true prime factorization is 2 × 2.”
The point is, even the best reflection process you can think of may fail to account for some crucial information. And hopefully it can robustly tell the difference between helpful information and harmful information.
For example, maybe N = 29522110801023785555247567907018022843013371193486904872915694135366948906267412459560469419313468477571904190875078325307783298702278061314706021273052523914864561727670955407896206738948955813504747448172831328073078012451035444606017289679166070717156612947440897221609673043263408054415375773691379283198201987372931659507826484639961297915624514954455314101431489726823065604374788650066472170603904794973458618994986833512839575283873771252517988691292017425081313700740089351712559811486464802367263467854576668024443015614104081670018991747427099820025784949521876071608490248395046666723258743709356296535782.
A = 1232477336227426692380443210888895449621471079543761850047395308187049594617001861601696813319110513848274876635355987350103099368741162622794253649965690823778078933544136385470959159681614005099869579024125795246308841648113743911773571553129957340479245748931756917137130013760495971399023845580343786885491244375889710319394039047959654983736068473827495579018934560487915085510788809201773576234050572295508350789032949910040027280393874600876759921821682717244338109892239475602768230768732937311567202612318055134309177705320521639635449240702247038285504267829595346064686009110312205007737419845285089599211
B = 28404358936141244817111713617600331891793591357912402403090074101183255558265948779426267547155101307334436867856129404340927293405086432682892248263827662441988227069219008681660301942027627323842312874031058512179223840896157038194006799237133125684806392915413774710883917390425119272589884765032741213943
C = 43390429581540126017572442049413488959139702613459795717022804047186295570874599744101057201841640892276368639418940532012449598478611604251554159646145486573777182692131295315212900911195947383450995718772416566366046388412317400464871279622896378528747304275985094440653456156567729772192224995294814959277