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.
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.
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.
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”.
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”.
You’re right, of course. To a large extent my comment sprung from a dislike of the idea that mathematics possesses some special ontological status independent of its relevance to our world—your point that even those statements which are parochial can be translated into terms comprehensible in a language fitted to a different sort of universe pretty much refutes that concern of mine.
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.
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”.
You’re right, of course. To a large extent my comment sprung from a dislike of the idea that mathematics possesses some special ontological status independent of its relevance to our world—your point that even those statements which are parochial can be translated into terms comprehensible in a language fitted to a different sort of universe pretty much refutes that concern of mine.