Huh. That’s interesting. Are you saying that you can actually pin down The Natural Numbers exactly using some “first order logic with branching quantifiers”? If so, I would be interested in seeing it.
dankane
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I am admittedly in a little out of my depth here, so the following could reasonably be wrong, but I believe that the Compactness Theorem can be proved within first order set theory. Given a consistent theory, I can use the axiom of choice to extend it to a maximal consistent set of statements (i.e. so that for every P either P or (not P) is in my set). Then for every statement that I have of the form “there exists x such that P(x)”, I introduce an element x to my model and add P(x) to my list of true statements. I then re-extend to a maximal set of statements, and add new variables as necessary, until I cannot do this any longer. What I am left with is a model for my theory. I don’t think I invoked second order logic anywhere here. In particular, what I did amounts to a construction within set theory. I suppose it is the case that some set theories will have no models of set theory (because they prove that set theory is inconsistent), while others will contain infinitely many.
My intuition on the matter is that if you can state what you are trying to say without second order logic, you should be able to prove it without second order logic. You need second order logic to even make sense of the idea of the standard natural numbers. The Compactness Theorem can be stated in first order set theory, so I expect the proof to be formalizable within first order set theory.
I’m not entirely sure what you’re getting at here. If we start restricting properties to only cut out sets of numbers rather than arbitrary collections, then we’ve already given up on full semantics.
If we take this leap, then it is a theorem of set theory that all set-theoretic models of the of the natural numbers are isomorphic. On the other hand, since not all statements about the integers can be either proven or disproven with the axioms of set theory, there must be different models of set theory which have different models of the integers within them (in fact, I can build these two models within a larger set theory).
On the other hand, if we continue to use full semantics, I’m not sure how you clarify to be what you mean when you say “a property exists for every collection of numbers”. Telling me that I should already know what a collection is doesn’t seem much more reasonable than telling me that I should already know what a natural number is.
OK then. As soon as you can explain to me exactly what you mean when you say “for any collection of numbers there is a corresponding property being quantified over”, I will be satisfied. In particular, what do you mean when you say “any collection”?
I am questioning the idea that numbers (at least the things that this post refers to as numbers) are a quality about objects. Numbers, as they are described here, are an abstract logical construction.
I don’t think that I understand what you mean here.
How can these properties represent causal relations? They are things that are satisfied by some numbers and not by others. Since numbers are aphysical, how do we relate this to causal relations.
On the other hand, even with a satisfactory answer to the above question, how do we know that “being in the first chain” is actually a property, since otherwise we still can’t show that there is only one chain.
Your idea of pinning down the natural numbers using second order logic is interesting, but I don’t think that it really solves the problem. In particular, it shouldn’t be enough to convince a formalist that the two of you are talking about the same natural numbers.
Even in second order PA, there will still be statements that are independent of the axioms, like “there doesn’t exist a number corresponding to a Godel encoding of a proof that 0=S0 under the axioms of second order PA”. Thus unless you are assuming full semantics (i.e. that for any collection of numbers there is a corresponding property), there should be distinct models of second order PA for which the veracity of the above statement differs.
Thus it seems to me that all you have done with your appeal to second order logic is to change my questions about “what is a number?” into questions about “what is a property?” In any case, I’m still not totally convinced that it is possible to pin down The Natural Numbers exactly.
I agree that mathematical truths do not have effects on their own. But when combined with mathematical formulations of laws of reality they do have observable consequences. The timing of a falling projectile above is a consequence of both a mathematical formulation of the law of gravity and a purely mathematical arithmetical fact. If you somehow want to describe the universe without mathematics, good luck.
Well I know that when I drop something the distance it falls after time t is roughly 1⁄2 g t^2 where g~10 m/s^2. When I drop something off of a 20m high building, I can reasonably claim that the fact that it takes roughly 2s to reach the ground is a consequence of the above, and of the mathematical truth that 1⁄2 10 2^2 = 20.
“For a statement to be comparable to your universe, so that it can be true or alternatively false, it must talk about stuff you can find in relation to yourself by tracing out causal links.”
With the above solution, then yes epiphenomenalist theories of consciousness are meaningful. They clearly describe networks in which consciousness connected to the rest of reality that you experience. On the other hand, I think that this is because the above statement fails at its intended goal rather than that these theories are actually meaningful. In particular I think one should append to the above:
“Furthermore, it must not be possible to remove this stuff (and perhaps some other stuff) from your causal network without affecting things that you can experience.”
Well, my experiences could not be readily predicted by a simple Bayesian net. Thus I would expect the joint probability distribution on all things to be really complicated unless there were some other kind of pattern to it. Heck, maybe my experiences correspond to repeated independent samples from a giant multidimensional Gaussian or something (rotated of course so that the natural observable variables have non-trivial covariance).
This seems especially difficult noting that although we can claim that things are caused by certain mathematical truths, it doesn’t really make sense to include them in our Bayesian net unless we could say, for example, how anything else would be different if 2+2=3.
If this is the case, then I’m confused as to what you mean by “true”. Let’s consider the statement “In the standard initial configuration in chess, there’s a helpmate in 2″. I imagine that you consider this analogous to your example of a statement about chess, but I am more comfortable with this one because it’s not clear exactly what a “poor move” is.
Now, if we wanted to explain this statement to a being from another universe, we would need to taboo “chess” and “helpmate” (and maybe “move”). The statement then unfolds into the following:
”In the game with the following set of rules… there is a sequence of play that causes the game to end after only two turns are taken by each player”
Now this statement is equivalent to the first, but seems to me like it is only more meaningful to us than it is to anyone else because the game it describes matches a game that we, in a universe where chess is well known, have a non-trivial probability of ever playing. It seems like you want to use “true” to mean “true and useful”, but I don’t think that this agrees with what most people mean by “true”.For example, there are infinitely many true statements of the form “A+B=C” for some specific integers A,B,C. On the other hand, if you pick A and B to be random really large numbers, the probability that the statement in question will ever be useful to anyone becomes negligible. On the other hand, it seems weird to start calling these statements “false” or “meaningless”.
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.
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.
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 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.
OK. I think that I had been misreading some of your previous posts. Allow me the rephrase my objection.
Suppose that our beliefs about photons were rewritten as “photons not beyond an event horizon obey Maxwell’s Equations”. Making this change to my belief structure now leaves beliefs about whether or not photons still exist beyond an event horizon unconnected from my experiences. Does the meaningfulness of this belief depend on how I phrase my other beliefs?
Also if one can equally easily produce belief systems which predict the same sets of experiences but disagree on whether or not the photon exists beyond the event horizon, how does this belief differ from the belief that Carol is a post-utopian?
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.
I don’t see how you can claim that the belief that the photon continues to exist is a meaningful belief without also allowing the belief that the photon does not continue to exist to be a meaningful belief. Unless you do something along the lines of taking Kolmogorov complexity into account, these beliefs seem to be completely analogous to each other. Perhaps to phrase things more neutrally, we should be asking if the question “does the photon continue to exist?” is meaningful. On the one hand, you might want to say “no” because the outcome of the question is epiphenomenal. On the other hand, you would like this question to be meaningful since it may have behavioral implications.
Cool. I agree that this is potentially less problematic than the second order logic approach. But it does still manage to encode the idea of a function in it implicitly when it talks about “y depending only on x”, it essentially requires that y is a function of x, and if it’s unclear exactly which functions are allowed, you will have problems. I guess first order logic has this problem to some degree, but with alternating quantifiers, the functions that you might need to define seem closer to the type that should necessarily exist.