How to Convince Me That 2 + 2 = 3

In “What is Ev­i­dence?” I wrote:1

This is why ra­tio­nal­ists put such a heavy pre­mium on the para­dox­i­cal-seem­ing claim that a be­lief is only re­ally worth­while if you could, in prin­ci­ple, be per­suaded to be­lieve oth­er­wise. If your retina ended up in the same state re­gard­less of what light en­tered it, you would be blind . . . Hence the phrase, “blind faith.” If what you be­lieve doesn’t de­pend on what you see, you’ve been blinded as effec­tively as by pok­ing out your eye­balls.

Cihan Baran replied:2

I can not con­ceive of a situ­a­tion that would make 2 + 2 = 4 false. Per­haps for that rea­son, my be­lief in 2 + 2 = 4 is un­con­di­tional.

I ad­mit, I can­not con­ceive of a “situ­a­tion” that would make 2 + 2 = 4 false. (There are re­defi­ni­tions, but those are not “situ­a­tions,” and then you’re no longer talk­ing about 2, 4, =, or +.) But that doesn’t make my be­lief un­con­di­tional. I find it quite easy to imag­ine a situ­a­tion which would con­vince me that 2 + 2 = 3.

Sup­pose I got up one morn­ing, and took out two earplugs, and set them down next to two other earplugs on my night­table, and no­ticed that there were now three earplugs, with­out any earplugs hav­ing ap­peared or dis­ap­peared—in con­trast to my stored mem­ory that 2 + 2 was sup­posed to equal 4. More­over, when I vi­su­al­ized the pro­cess in my own mind, it seemed that mak­ing xx and xx come out to xxxx re­quired an ex­tra x to ap­pear from nowhere, and was, more­over, in­con­sis­tent with other ar­ith­metic I vi­su­al­ized, since sub­tract­ing xx from xxx left xx, but sub­tract­ing xx from xxxx left xxx. This would con­flict with my stored mem­ory that 3 − 2 = 1, but mem­ory would be ab­surd in the face of phys­i­cal and men­tal con­fir­ma­tion that xxxxx = xx.

I would also check a pocket calcu­la­tor, Google, and per­haps my copy of 1984 where Win­ston writes that “Free­dom is the free­dom to say two plus two equals three.” All of these would nat­u­rally show that the rest of the world agreed with my cur­rent vi­su­al­iza­tion, and dis­agreed with my mem­ory, that 2 + 2 = 3.

How could I pos­si­bly have ever been so de­luded as to be­lieve that 2 + 2 = 4? Two ex­pla­na­tions would come to mind: First, a neu­rolog­i­cal fault (pos­si­bly caused by a sneeze) had made all the ad­di­tive sums in my stored mem­ory go up by one. Se­cond, some­one was mess­ing with me, by hyp­no­sis or by my be­ing a com­puter simu­la­tion. In the sec­ond case, I would think it more likely that they had messed with my ar­ith­metic re­call than that 2 + 2 ac­tu­ally equalled 4. Nei­ther of these plau­si­ble-sound­ing ex­pla­na­tions would pre­vent me from notic­ing that I was very, very, very con­fused.3

What would con­vince me that 2 + 2 = 3, in other words, is ex­actly the same kind of ev­i­dence that cur­rently con­vinces me that 2 + 2 = 4: The ev­i­den­tial cross­fire of phys­i­cal ob­ser­va­tion, men­tal vi­su­al­iza­tion, and so­cial agree­ment.

There was a time when I had no idea that 2 + 2 = 4. I did not ar­rive at this new be­lief by ran­dom pro­cesses—then there would have been no par­tic­u­lar rea­son for my brain to end up stor­ing “2 + 2 = 4” in­stead of “2 + 2 = 7.” The fact that my brain stores an an­swer sur­pris­ingly similar to what hap­pens when I lay down two earplugs alongside two earplugs, calls forth an ex­pla­na­tion of what en­tan­gle­ment pro­duces this strange mir­ror­ing of mind and re­al­ity.

There’s re­ally only two pos­si­bil­ities, for a be­lief of fact—ei­ther the be­lief got there via a mind-re­al­ity en­tan­gling pro­cess, or not. If not, the be­lief can’t be cor­rect ex­cept by co­in­ci­dence. For be­liefs with the slight­est shred of in­ter­nal com­plex­ity (re­quiring a com­puter pro­gram of more than 10 bits to simu­late), the space of pos­si­bil­ities is large enough that co­in­ci­dence van­ishes.4

Un­con­di­tional facts are not the same as un­con­di­tional be­liefs. If en­tan­gled ev­i­dence con­vinces me that a fact is un­con­di­tional, this doesn’t mean I always be­lieved in the fact with­out need of en­tan­gled ev­i­dence.

I be­lieve that 2 + 2 = 4, and I find it quite easy to con­ceive of a situ­a­tion which would con­vince me that 2 + 2 = 3. Namely, the same sort of situ­a­tion that cur­rently con­vinces me that 2 + 2 = 4. Thus I do not fear that I am a vic­tim of blind faith.5

1See Map and Ter­ri­tory.

2Com­ment: http://​​less­wrong.com/​​lw/​​jl/​​what_is_ev­i­dence/​​f7h.

3See “Your Strength as a Ra­tion­al­ist” in Map and Ter­ri­tory.

4For more on be­lief for­ma­tion and be­liefs of fact, see “Feel­ing Ra­tional” and “What Is Ev­i­dence?” in Map and Ter­ri­tory. For more on be­lief com­plex­ity, see “Oc­cam’s Ra­zor” in the same vol­ume.

5If there are any Chris­ti­ans read­ing this who know Bayes’s The­o­rem, might I in­quire of you what situ­a­tion would con­vince you of the truth of Is­lam? Pre­sum­ably it would be the same sort of situ­a­tion causally re­spon­si­ble for pro­duc­ing your cur­rent be­lief in Chris­ti­an­ity: We would push you scream­ing out of the uterus of a Mus­lim woman, and have you raised by Mus­lim par­ents who con­tinu­ally told you that it is good to be­lieve un­con­di­tion­ally in Is­lam.

Or is there more to it than that? If so, what situ­a­tion would con­vince you of Is­lam, or at least, non-Chris­ti­an­ity? And how con­fi­dent are you that the gen­eral kinds of ev­i­dence and rea­son­ing you ap­peal to would have been enough to dis­suade you of your re­li­gion if you had been raised a Mus­lim?