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orthonormal comments on How to Convince Me That 2 + 2 = 3 - Less Wrong

53 Post author: Eliezer_Yudkowsky 27 September 2007 11:00PM

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Comment author: orthonormal 31 July 2009 01:21:12AM 8 points [-]

It's often poor form to quote oneself, but since this post (deservedly) continues to get visits, it might be good to bring up the line of thought that convinced me that this post made perfect sense:

The space of all possible minds includes some (aliens/mental patients/AIs) which have a notion of number and counting and an intuitive mental arithmetic, but where the last of these is skewed so that 2 and 2 really do seem to make 3 rather than 4. Not just lexically, but actually; the way that our brains can instantly subitize four objects as two distinct groups of two, their minds mistakenly "see" the pattern 0 0 0 as composed of two distinct 0 0 groups. Although such a mind would be unlikely to arise within natural selection, there's nothing impossible about engineering a mind with this error, or rewiring a mind within a simulation to have this error.

These minds, of course, would notice empirical contradictions everywhere: they would put two objects together with two more, count them, and then count four instead of three, when it's obvious by visualizing in their heads that two and two ought to make three instead. They would even encounter proofs that 2 + 2 =4, and be unable to find an error, although it's patently absurd to write SSSS0 = SS0 + SS0. Eventually, a sufficiently reflective and rational mind of this type might entertain the possibility that maybe two and two do actually make four, and that its system of visualization and mental arithmetic are in fact wrong, as obvious as they seem from the inside. We would consider such a mind to be more rational than one that decided that, no matter what it encountered, it could never be convinced that 2 and 2 made 4 rather than 3.

Now, given all that, why exactly should I refuse to ever update my arithmetical beliefs if given the sort of experiences in Eliezer's thought experiment? Wouldn't the hypothesis that I am such an agent get a lot of confirmation? (Of course, I very strongly don't expect to encounter such experiences, because of all the continuing evidence before me that 2 + 2 = 4; but if I did wake up in that situation, I'd have to accept that some part of my mind is probably broken, and the part that tells me 2 + 2 = 4 is as likely a candidate as any.)

Comment author: byrnema 31 July 2009 03:13:18AM *  1 point [-]

I had parallel thoughts at one time, and discovered with some effort that I could train myself to believe that 1+1=3. It took about five minutes of mental practice. What eventually happened was that every time I combined two objects together mentally (abstractly), I simultaneously imagined a third which had the bizarre property that it only existed when the two objects were considered simultaneously. If I thought of just one object, the third disappeared, if I thought of the other object, it again disappeared -- it only appeared as an emergent property of the pair. Thus imagining 1+1=3 was discovering the following "operation":

{E} + {F} = { {E} , {F} , { {E},{F} } }

Looking at the cardinality of the sets, we have: 1 + 1 = 3

Could such an operation be 'logical' and yield a consistent number theory? (I don't know. I think it's a question in abstract algebra. (Rings, fields, groups, etc.) Are there any algebraists here that can comment?)

Yet orthonormal is suggesting the case that 2+2=3 doesn't result in a logical, consistent theory -- the possible minds just believe it due to an internal error, and they can use the inconsistency of their theory to deduce the internal error. However, I find it really difficult to think of 2+2=3 happening as a mistaken Peano arithmetic instead of the assertion of another type of arithmetic. The possible logical self-consistency of this arithmetic further confounds: if it's self-consistent, they may never deduce that they got Peano arithmetic wrong. If its not self-consistent, they can prove all propositions and how will they know where the error lies? Or even understand what error means? If there is an error in our reasoning, it cannot be so fundamentally embedded in our understanding of logic.

Comment author: danilobellini 02 August 2009 07:48:30AM *  -1 points [-]

Nice, but the difference with this "belief" is that you're talking about sensory "counting" (visual grouping), and I was talking about the numbers themselves, as models for games, other phenomena, etc., and not just as a "counting" tool.

In the 1+1=3 example, to define the cardinality, he/she used the Peano's axioms, didn't he/she?

I don't see the "visual sensory counting" as the only use for "2+2=4", that's why I don't think this experiment would refute such a priori content.

Another idea: let Ann be a girl with hemispatial neglect in a extinction condition. Ann has problems detecting anything on the left, and she can possibly see 2+2=3 as idealized above, due her brain damage. Will she think that 2+2=3? I don't think so...but if she does...will that be a model for all "integer numbers" aplications? I think in "integer" as a framework for several phenomena, other models, other knowledge, not only the counting one.

For the minds that see 2+2=4 as something patently absurd, because 2+2=3 is part of their intuitive arithmetic, these minds probably won't see the 2+2=4 even when brought to a world like ours. After a time in the 2+2=4 world, they probably won't forget that 2+2=3, unless the 2+2=3 wasn't modeling anything else. But the 2+2=3 was modeling something in their past history, at least the counting principle of their world. So they still have the 2+2=3 belief in their lives while they remember their past. If they forget their past, the 2+2=3 belief might became unuseful, but that still don't make the 2+2=3 an absurd or replaced by the 2+2=4: there are 2 number systems here.

For me, 2+2=3 isn't an absurd. That might be seem as a "common sum with a 3/4 multiplier" or a "X + Y = X p Y/X" where "p" is our common sum and "/" is our division, etc.. This way, like the 1+1=3 example above, only overloads the "+" operator. But, again, this "+" isn't the same from the "2+2=4"

Comment author: Kingreaper 26 November 2010 01:39:33AM 0 points [-]

Upon suddenly discovering that the whole world looks different this morning than it did last night is the rational belief "I guess I was deluded for my whole life up to this point" or "I guess I'm deluded now"?

Considering the fact that you're not waking up in a mental institution, but the world still seems to contain them (and if you get 2 sets of 2 of them, you have 3); the latter is a much more likely situation

Comment author: orthonormal 27 November 2010 10:57:43PM 0 points [-]

You're neglecting the hypothesis "my memories of the past are being distorted to convince me that 2 and 2 make 4 instead of 3". Given how easily we distort our memories under conditions of sanity, this is as likely as "I'm deluded now".

Comment author: Kingreaper 28 November 2010 12:08:32PM *  0 points [-]

If you suddenly gain a set of memories indicating that the raptor conspiracy is taking over the world, you would be considered deluded.

If you suddenly gain a set of memories indicating that 2+2 equals something other than what it DOES in fact equal, you are likewise deluded.

So your suggestion is in fact a subset of being deluded*. At which point you should voluntarily seek out psychological/psychiatric help.

  • (which I assign a low probability, as I have never heard of such a type of delusion existing)

If you believe (as you seem to suggest by use of the aactive rather than the passive voice) that this delusion is being deliberately induced, it is important to remember that anyone with the power to induce that delusion could also reduce you to a gibbering wreck; and hence that going to get help is highly unlikely to be "part of their plan".

Comment author: orthonormal 28 November 2010 08:58:11PM 0 points [-]

This is a distraction from the actual point; of course if this happened to me, then my first priority would be getting help (I might be having a stroke, for instance). But once I'm at the hospital and they tell me that I'm all right, but something strange happened to my brain so that it falsely remembers 2 and 2 having made 4, instead of the obviously correct 3...

If you don't agree that some set of circumstances like this should conspire to make me rationally accept 2+2=3, then if the scenario happened to you (with 3 and 4 reversed), you're asserting that you could never rationally recover from that metal event. Since I'd prefer, should I go through a hallucination that 2 and 2 always made 3, to be able to recover given enough evidence, I have to take the "risk" of being convinced of something false, in a world where events conspired against me just so.

Comment author: wedrifid 28 November 2010 12:21:16PM *  2 points [-]

Upon suddenly discovering that the whole world looks different this morning than it did last night is the rational belief "I guess I was deluded for my whole life up to this point" or "I guess I'm deluded now"?

Why completely leave out the possibility that you aren't deluded at all? Depending on just what kind of 'different' you wake up in that is a distinct possibility.

I would, by the way, start with a high prior for 'deluded now' which would be altered one way or the other by extensive reality testing. I experience that in dreams all the time. I know from personal experience it is easier for me to be confused about the transient sensory experience of the present than the broad structure of all my memories. Results may vary somewhat.

Comment author: Kingreaper 28 November 2010 01:55:03PM *  0 points [-]

Good point, in the case of waking up in a logically possible world, remembering a previous logically possible world, there is a non-zero possibility that you've actually gone from one to the other somehow. How low the probability is depends on the nature of the differences

I was too caught up in the case of waking up in a world where the world you remember is logically impossible.

Comment author: wedrifid 29 November 2010 12:27:23AM *  0 points [-]

Good point, in the case of waking up in a logically possible world, remembering a previous logically possible world, there is a non-zero possibility that you've actually gone from one to the other somehow. How low the probability is depends on the nature of the differences

Exactly. And with slightly different wording a world in which it seems like you have changed from one logical world to another is itself a just a logically possible world.

I was too caught up in the case of waking up in a world where the world you remember is logically impossible.

That would be awkward! It would require an awful lot of reality testing on the question of just how logically impossible things were. Even after that your confidence in just about anything would be fubared.