Less Wrong is a community blog devoted to refining the art of human rationality. Please visit our About page for more information.

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

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

You are viewing a comment permalink. View the original post to see all comments and the full post content.

Comments (390)

Sort By: Old

You are viewing a single comment's thread. Show more comments above.

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"