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khafra comments on The Useful Idea of Truth - Less Wrong
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So my belief that 2+2=4 isn't meaningful?
I thought Eliezer's story about waking up in a universe where 2+2 seems to equal 3 felt pretty coherent.
edit: It seems like the story would be less coherent if it involved detailed descriptions of re-deriving mathematics from first principles. So perhaps ArisKatsaris' definition leaves too much to the author's judgement in what to leave out of the story.
I think that it's a good deal more subtle than this. Eliezer described a universe in which he had evidence that 2+2=3, not a universe in which 2 plus 2 was actually equal to 3. If we talk about the mathematical statement that 2+2=4, there is actually no universe in which this can be false. On the other hand in order to know this fact we need to acquire evidence of it, which, because it is a mathematical truth, we can do without any interaction with the outside world. On the other hand if someone messed with your head, you could acquire evidence that 2 plus 2 was 3 instead, but seeing this evidence would not cause 2 plus 2 to actually equal 3.
On the contrary. Imagine a being that cannot (due to some neurological quirk) directly percieve objects - it can only percieve the spaces between objects, and thus indirectly deduce the presence of the objects themselves. To this being, the important thing - the thing that needs to be counted and to which a number is assigned - is the space, not the object.
Thus, "two" looks like this, with two spaces: 0 0 0
Placing "two" next to "two" gives this: 0 0 0 0 0 0
Counting the spaces gives five. Thus, 2+2=5.
I think you misunderstand what I mean by "2+2=4". Your argument would be reasonable if I had meant "when you put two things next to another two things I end up with four things". On the other hand, this is not what I mean. In order to get that statement you need the additional, and definitely falsifiable statement "when I put a things next to b things, I have a+b things".
When I say "2+2=4", I mean that in the totally abstract object known as the natural numbers, the identity 2+2=4 holds. On the other hand the Platonist view of mathematics is perhaps a little shaky, especially among this crowd of empiricists, so if you don't want to accept the above meaning, I at least mean that "SS0+SS0=SSSS0" is a theorem in Peano Arithmetic. Neither of these claims can be false in any universe.
I think I understand what CCC means by the being that perceives spaces instead of objects - Peano Arithmetic only exists because it is useful for us, human beings, to manipulate numbers that way. Given a different set of conditions, a different set of mathematical axioms would be employed.
Peano Arithmetic is merely a collection of axioms (and axiom schema), and inference laws. It's existence is not predicated upon its usefulness, and neither are its theorems.
I agree that the fact that we actually talk about Peano Arithmetic is a consequence of the fact that it (a) is useful to us (b) appeals to our aesthetic sense. On the other hand, although the being described in CCC's post may not have developed Peano's axioms on their own, once they are informed of these axioms (and modus ponens, and what it means for something to be a theorem), they would still agree that "SS0+SS0=SSSS0" in Peano Arithmetic.
In summary, although there may be universes in which the belief "2+2=4" is no longer useful, there are no universes in which it is not true.
I freely concede that a tree falling in the woods with no-one around makes acoustic vibrations, but I think it is relevant that it does not make any auditory experiences.
In retrospect, however, backtracking to the original comment, if "2+2=4" were replaced by "not(A and B) = (not A) or (not B)", I think my argument would be nearly untenable. I think that probably suffices to demonstrate that ArisKatsaris's theory of meaningfulness is flawed.
How is it relevant? CCC was arguing that "2+2=4" was not true in some universes, not that it wouldn't be discovered or useful in all universes. If your other example makes you happy that's fine, but I think it would be possible to find hypothetical observers to whom De Morgan's Law is equally useless. For example, the observer trapped in a sensory deprivation chamber may not have enough in the way of actual experiences for De Morgan's Law to be at all useful in making sense of them.
In my opinion, saying "2+2=4 in every universe" is roughly equivalent to saying "1.f3 is a poor chess opening in every universe" - it's "true" only if you stipulate a set of axioms whose meaningfulness is contingent on facts about our universe. It's a valid interpretation of the term "true", but it is not the only such interpretation, and it is not my preferred interpretation. That's all.