(There appear to be two useful senses of “doesn’t exist”: the state of some system is such that some property isn’t present; or a description of a system is contradictory. These don’t obviously apply here.)
In set theory there’s a formal symbol that’s read “there exists”, and a pair of formal symbols that are read “there doesn’t exist”. Do you think these symbols should be understood in either of your two useful senses?
It is possible to read the existential quantifier as “for some” instead of “there exists … such that”. I often do this myself, just for euphony (and to match the dual quantifier, read “for all”, or better “for each”). But Graham Priest (pdf) has argued that the “there exists” reading is a case of ontological sleight of hand that should be resisted; in fact, he rejects the term “existential quantifier” for “particular quantifier” (and a web search for this will turn up more on the subject).
Priest (top of page 3 in the PDF above, numbered page 199) suggests an example:
I thought of something I would like to buy you for Christmas, but I couldn’t get it because it doesn’t exist.
In symbols:
∃ x, (I thought of x) & (I would like to buy you x for Christmas) & [(I couldn’t get x) ∵ (x doesn’t exist)].
Turning this back into English:
For some x, I thought of x, I would like to buy you x for Christmas, and I couldn’t get x because x doesn’t exist.
But not this:
There exists x such that I thought of x, I would like to buy you x for Christmas, and I couldn’t get x because x doesn’t exist.
One could rescue this by claiming that x exists in the speaker’s past thoughts but not in reality, or something like that. But then an uncountable ordinal may also exist in the thoughts of mathematicians without existing in reality.
There are many models of interest in set theory, with different mutually exclusive properties. A logical statement makes sense in context of axioms or intended model. I didn’t take Eliezer’s comment as referring to either of these technical senses (it’s not expecting provability of nonexistence from standard axiom systems, since they just assert existence in question, the alternative being asserting inconsistency, which would be easier to state directly; and standard model is an unclear proposition for set theory, there being so many alternatives, with one taken as the usual standard containing the elements in question). So I was talking about “ontological” senses of “existence” instead.
ZF proves in formal symbols “there exists a smallest uncountable ordinal,” and I guess you are saying that it does not mean that in an ontological sense. But then what is the ontological payoff of this proof?
Like any proof in a formal system, you can conclude that “the idea is consistent unless the formal system is inconsistent.” But that’s a tautology. If you’re not willing to say that ZF refers to things in the real world i.e. has ontological content, why aren’t you skeptical of it?
Like any proof in a formal system, you can conclude that “the idea is consistent unless the formal system is inconsistent.” But that’s a tautology.
I wasn’t saying that. If you believe that a formal system captures the idea you’re considering, in the sense of this idea being about properties of (some of) the models of this formal system, and the formal system tells you that the idea doesn’t make sense, it’s some evidence towards the idea not making sense, even though it’s also possible that the formal system is just broken, or that it doesn’t actually capture the idea, and you need to look for a different formal system to perceive it properly.
If you’re not willing to say that ZF refers to things in the real world i.e. has ontological content, why aren’t you skeptical of it?
ZF clearly refers to lots of things not related to the physical world, but if it’s not broken (and it doesn’t look like it is), it can talk about many relevant ideas, and help in answering questions about these ideas. It can tell whether some object doesn’t hold some property, for example, or whether some specification is contradictory.
(I know a better term for my current philosophy of ontology now: “mathematical monism”. From this POV, inference systems are just another kind of abstract object, as is their physical implementation in mathematicians’ brains. Inference systems are versatile tools for “perceiving” other facts, in the sense that (some of) the properties of those other facts get reflected as the properties of the inference systems, and consequently as the properties of physical devices implementing or simulating the inference systems. An inference system may be unable to pinpoint any one model of interest, but it still reflects its properties, which is why failure to focus of a particular model or describe what it is, is not automatically a failure to perceive some properties of that model. Morality is perhaps undefinable in this sense.)
This is again not the sense I discussed. A claim that an uncountable ordinal “doesn’t exist” has to be interpreted in a different way to make any sense. A claim that it does doesn’t need such excursions, and so the default senses of these claims are unrelated.
In set theory there’s a formal symbol that’s read “there exists”, and a pair of formal symbols that are read “there doesn’t exist”. Do you think these symbols should be understood in either of your two useful senses?
It is possible to read the existential quantifier as “for some” instead of “there exists … such that”. I often do this myself, just for euphony (and to match the dual quantifier, read “for all”, or better “for each”). But Graham Priest (pdf) has argued that the “there exists” reading is a case of ontological sleight of hand that should be resisted; in fact, he rejects the term “existential quantifier” for “particular quantifier” (and a web search for this will turn up more on the subject).
I can’t think of a situation where I would accept one but not the other of “there exists x such that—” and “for some x—”. Do you have an example?
Godel has a very interesting paper about syntax for intuitionism, where he introduces a new operator read “there exists constructively.”
Priest (top of page 3 in the PDF above, numbered page 199) suggests an example:
In symbols:
Turning this back into English:
But not this:
One could rescue this by claiming that x exists in the speaker’s past thoughts but not in reality, or something like that. But then an uncountable ordinal may also exist in the thoughts of mathematicians without existing in reality.
There are many models of interest in set theory, with different mutually exclusive properties. A logical statement makes sense in context of axioms or intended model. I didn’t take Eliezer’s comment as referring to either of these technical senses (it’s not expecting provability of nonexistence from standard axiom systems, since they just assert existence in question, the alternative being asserting inconsistency, which would be easier to state directly; and standard model is an unclear proposition for set theory, there being so many alternatives, with one taken as the usual standard containing the elements in question). So I was talking about “ontological” senses of “existence” instead.
ZF proves in formal symbols “there exists a smallest uncountable ordinal,” and I guess you are saying that it does not mean that in an ontological sense. But then what is the ontological payoff of this proof?
Don’t know, maybe not denying that the idea is consistent, and so doesn’t “doesn’t exist” in that sense?
Like any proof in a formal system, you can conclude that “the idea is consistent unless the formal system is inconsistent.” But that’s a tautology. If you’re not willing to say that ZF refers to things in the real world i.e. has ontological content, why aren’t you skeptical of it?
I wasn’t saying that. If you believe that a formal system captures the idea you’re considering, in the sense of this idea being about properties of (some of) the models of this formal system, and the formal system tells you that the idea doesn’t make sense, it’s some evidence towards the idea not making sense, even though it’s also possible that the formal system is just broken, or that it doesn’t actually capture the idea, and you need to look for a different formal system to perceive it properly.
ZF clearly refers to lots of things not related to the physical world, but if it’s not broken (and it doesn’t look like it is), it can talk about many relevant ideas, and help in answering questions about these ideas. It can tell whether some object doesn’t hold some property, for example, or whether some specification is contradictory.
(I know a better term for my current philosophy of ontology now: “mathematical monism”. From this POV, inference systems are just another kind of abstract object, as is their physical implementation in mathematicians’ brains. Inference systems are versatile tools for “perceiving” other facts, in the sense that (some of) the properties of those other facts get reflected as the properties of the inference systems, and consequently as the properties of physical devices implementing or simulating the inference systems. An inference system may be unable to pinpoint any one model of interest, but it still reflects its properties, which is why failure to focus of a particular model or describe what it is, is not automatically a failure to perceive some properties of that model. Morality is perhaps undefinable in this sense.)
This is again not the sense I discussed. A claim that an uncountable ordinal “doesn’t exist” has to be interpreted in a different way to make any sense. A claim that it does doesn’t need such excursions, and so the default senses of these claims are unrelated.