I think if you think anything has a perfect property, then you believe that perfect-essential necessity is realistic, basically. If you disagree that anything has a perfect property, then yeah, I think that’s pretty reasonable.
Couldn’t I think the property of “being logically consistent” is perfect, and possible, without thinking “perfect-essential necessity” is perfect (and therefore possible)?
No, because then there exhibits something with perfect-essential necessity, right? So it’s coherent to call it perfect, because it is possibly realized. As long as being logically consistent is a perfect-generating property of… something. An axiomatic system perhaps?
In this case I haven’t accepted that perfect-essential necessity is a perfect property. Peano arithmetic, for example, is (I claim) logically consistent, but does not have the property of perfect-essential necessity. And suppose I claim logical consistency is a perfect property. This seems ok so far.
Ah sure, so do I have you correct in saying that you think there is no essence that generates all the perfect properties of Peano Arithmetic? (Could I get around this by just taking the conjunction of all of its perfect properties, and saying that this is the perfect-generating property? It would satisfy the definition of a perfect-generating property, assuming you think that the conjunction necessarily implies each of the individual perfect properties in every world, which I think is not too controversial of a claim).
Idk. But suppose there is an essence that generates all perfect properties of PA. How would “perfect-essential necessity is possible” follow? (And if it does follow, why do you need axiom 6?)
Peano Arithmetic has a perfect-generating property.
It is possible for that perfect-generating property to be (EDIT: necessarily) instantiated (it is instantiated in Peano Arithmetic).
Therefore it is possible for something to have the property of perfect-essential necessity.
To your other point, we are making the claim that perfect-essential necessity is perfect. I think “perfect-essential necessity is possible” is a bit weaker, but axiom 6 does also claim that. I thought we were discussing whether axiom 6 is strange because you think nothing has the property of perfect-essential necessity, which would be required to be able to call it perfect.
EDIT: No hostility intended, by the way, I’m just wanting to be clear that I’m addressing the correct point.
“PA has a perfect generating property” let is suppose this. for example, maybe there is some list of perfect properties of PA, and they all follow from some property of it.
“It is possible for that perfect-generating property to be instantiated (it is instantiated in Peano Arithmetic)” sure, ok
“Therefore it is possible for something to have the property of perfect-essential necessity.” absolutely wild claim.
“We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.”
Seems like a wild claim. For PA to have this property would mean that its perfect-generating property (perhaps logical consistency, or soundness?) has a corresponding object having that property (e.g. soundness) in every possible world. But that’s a wild claim. Maybe there’s a possible universe with so little information that no sound logical system exists there.
You think there are logically possible worlds without sound logical systems? I guess it depends on your interpretation of modal logic, but I interpret it as “logical necessity” or “logical contingency,” so we aren’t like breaking the rules of logic or anything...
EDIT: I will probably also defend this interpretation reasonably strongly.
Imagine a universe that consists of a single particle. The law is that the particle stays in the same state always. Actually, the particle only has 1 possible state in this universe. So it’s just always in the same state as a matter of math not just physics.
Okay, noting that you made a stronger claim, that there is a universe where there is no sound logical system (i.e. P and not P are true in some possible universe), but I’m happy to move to this, sure.
What do you think it is about a Universe that is “making PA exist.” If tomorrow physicists found out there are <=10^googol particles in the universe, and 2^(10^googol) possible arrangements of states of the universe, or some other laughably huge number, would you then say “PA doesn’t exist in this universe?” Because PA proves 2^(10^googol) + 1 exists? I’m unclear about the ontology you’re assuming?
Let us say that PA consists of a formal system with exactly the standard axioms of PA. There is some quibbling about which “paper copies” work or which “translations” work. But PA only exists in a universe if it is instantiated as physical information. Instantiating large numbers does not instantiate PA.
Okay, noting that you made a stronger claim, that there is a universe where there is no sound logical system (i.e. P and not P are true in some possible universe), but I’m happy to move to this, sure.
Not what I meant. Two interpretations (a) in the single-particle universe, no logical system, sound or unsound, is instantiated, because no non-trivial physical info is instantiated. (b) Perhaps we could say that PA’s prefect-generating property is stronger than soundness. It is soundness plus “having provably distinct terms that number in the countable infinity”.
I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).
Let me move to a different example, why do you think our universe instantiates PA? Or do you not? Why do you think it has to be instantiated physically to exist? E.g. It can be proven that Con(ZFC) is true iff a certain turing machine with 745 states halts. This can be done in the metatheory of PA, I’m relatively confident (though not certain) Do you think that our universe then either instantiates Con(ZFC) or doesn’t? Since PA exists? Seems like a weird ontology for maths to me, but I guess all ontologies for maths are weird.
I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).
Nope, because they didn’t write down the system. (This might be quibbling. I mean it doesn’t write down that formal system anywhere.)
Let me move to a different example, why do you think our universe instantiates PA
Because mathematicians have written down the rules of PA. And they have programmed computers to use PA.
Do you think that our universe then either instantiates Con(ZFC) or doesn’t?
Our universe instantiates formal system such as ZFC + Con(ZFC), because mathematicians can write them down. (It also instantiates inconsistent formal systems such as “ZFC + axiom of choice is false”.)
Backtracking a bit. I reject axiom 6 (which is not to say I actively disbelieve it, but that I don’t actively believe it) and I’m giving an idea of why. If you could logically prove axiom 6, you wouldn’t need it to be an axiom. So you’re going to have some trouble convincing me of it on logical grounds. Rather it’s more metaphysical and so on.
Yes, yes yes, this is of course fine to reject the axiom I agree. But the metaphysics is where it actually gets good (I only did a maths degree and am leeching the metaphysics of it out of my system with posts like these lol).
I mean I think this is a plausibly reasonable account of when mathematical objects exist I guess, kind of a “structuralism” flavour to it, I’m actually somewhat sympathetic. I didn’t think of that objection!
Though I will note this is only one example of a potential object with perfect-essential necessity. I’ve linked the formalization which comes up with a logical model of these axioms (including all the necessity, perfect-essential necessity, etc) in a different thread on your original comment, if you’re curious!
If you’d like to stop back-and-forthing about metaphysics, seems reasonable. I’m sure we’ll make lots of progress if we keep going debating our priors about this! \s
I think if you think anything has a perfect property, then you believe that perfect-essential necessity is realistic, basically. If you disagree that anything has a perfect property, then yeah, I think that’s pretty reasonable.
Couldn’t I think the property of “being logically consistent” is perfect, and possible, without thinking “perfect-essential necessity” is perfect (and therefore possible)?
No, because then there exhibits something with perfect-essential necessity, right? So it’s coherent to call it perfect, because it is possibly realized. As long as being logically consistent is a perfect-generating property of… something. An axiomatic system perhaps?
In this case I haven’t accepted that perfect-essential necessity is a perfect property. Peano arithmetic, for example, is (I claim) logically consistent, but does not have the property of perfect-essential necessity. And suppose I claim logical consistency is a perfect property. This seems ok so far.
Ah sure, so do I have you correct in saying that you think there is no essence that generates all the perfect properties of Peano Arithmetic? (Could I get around this by just taking the conjunction of all of its perfect properties, and saying that this is the perfect-generating property? It would satisfy the definition of a perfect-generating property, assuming you think that the conjunction necessarily implies each of the individual perfect properties in every world, which I think is not too controversial of a claim).
Idk. But suppose there is an essence that generates all perfect properties of PA. How would “perfect-essential necessity is possible” follow? (And if it does follow, why do you need axiom 6?)
Peano Arithmetic is (EDIT: necessarily) possible.
Peano Arithmetic has a perfect-generating property.
It is possible for that perfect-generating property to be (EDIT: necessarily) instantiated (it is instantiated in Peano Arithmetic).
Therefore it is possible for something to have the property of perfect-essential necessity.
To your other point, we are making the claim that perfect-essential necessity is perfect. I think “perfect-essential necessity is possible” is a bit weaker, but axiom 6 does also claim that. I thought we were discussing whether axiom 6 is strange because you think nothing has the property of perfect-essential necessity, which would be required to be able to call it perfect.
EDIT: No hostility intended, by the way, I’m just wanting to be clear that I’m addressing the correct point.
“PA is possible” sure ok.
“PA has a perfect generating property” let is suppose this. for example, maybe there is some list of perfect properties of PA, and they all follow from some property of it.
“It is possible for that perfect-generating property to be instantiated (it is instantiated in Peano Arithmetic)” sure, ok
“Therefore it is possible for something to have the property of perfect-essential necessity.” absolutely wild claim.
“We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.”
Seems like a wild claim. For PA to have this property would mean that its perfect-generating property (perhaps logical consistency, or soundness?) has a corresponding object having that property (e.g. soundness) in every possible world. But that’s a wild claim. Maybe there’s a possible universe with so little information that no sound logical system exists there.
You think there are logically possible worlds without sound logical systems? I guess it depends on your interpretation of modal logic, but I interpret it as “logical necessity” or “logical contingency,” so we aren’t like breaking the rules of logic or anything...
EDIT: I will probably also defend this interpretation reasonably strongly.
Imagine a universe that consists of a single particle. The law is that the particle stays in the same state always. Actually, the particle only has 1 possible state in this universe. So it’s just always in the same state as a matter of math not just physics.
There is no way for PA to exist in this universe.
Okay, noting that you made a stronger claim, that there is a universe where there is no sound logical system (i.e. P and not P are true in some possible universe), but I’m happy to move to this, sure.
What do you think it is about a Universe that is “making PA exist.” If tomorrow physicists found out there are <=10^googol particles in the universe, and 2^(10^googol) possible arrangements of states of the universe, or some other laughably huge number, would you then say “PA doesn’t exist in this universe?” Because PA proves 2^(10^googol) + 1 exists? I’m unclear about the ontology you’re assuming?
Let us say that PA consists of a formal system with exactly the standard axioms of PA. There is some quibbling about which “paper copies” work or which “translations” work. But PA only exists in a universe if it is instantiated as physical information. Instantiating large numbers does not instantiate PA.
Not what I meant. Two interpretations (a) in the single-particle universe, no logical system, sound or unsound, is instantiated, because no non-trivial physical info is instantiated. (b) Perhaps we could say that PA’s prefect-generating property is stronger than soundness. It is soundness plus “having provably distinct terms that number in the countable infinity”.
I feel like the single particle universe clearly instantiates the logical system: ∃x (particle(x)).
Let me move to a different example, why do you think our universe instantiates PA? Or do you not? Why do you think it has to be instantiated physically to exist? E.g. It can be proven that Con(ZFC) is true iff a certain turing machine with 745 states halts. This can be done in the metatheory of PA, I’m relatively confident (though not certain) Do you think that our universe then either instantiates Con(ZFC) or doesn’t? Since PA exists? Seems like a weird ontology for maths to me, but I guess all ontologies for maths are weird.
Nope, because they didn’t write down the system. (This might be quibbling. I mean it doesn’t write down that formal system anywhere.)
Because mathematicians have written down the rules of PA. And they have programmed computers to use PA.
Our universe instantiates formal system such as ZFC + Con(ZFC), because mathematicians can write them down. (It also instantiates inconsistent formal systems such as “ZFC + axiom of choice is false”.)
Backtracking a bit. I reject axiom 6 (which is not to say I actively disbelieve it, but that I don’t actively believe it) and I’m giving an idea of why. If you could logically prove axiom 6, you wouldn’t need it to be an axiom. So you’re going to have some trouble convincing me of it on logical grounds. Rather it’s more metaphysical and so on.
Yes, yes yes, this is of course fine to reject the axiom I agree. But the metaphysics is where it actually gets good (I only did a maths degree and am leeching the metaphysics of it out of my system with posts like these lol).
I mean I think this is a plausibly reasonable account of when mathematical objects exist I guess, kind of a “structuralism” flavour to it, I’m actually somewhat sympathetic. I didn’t think of that objection!
Though I will note this is only one example of a potential object with perfect-essential necessity. I’ve linked the formalization which comes up with a logical model of these axioms (including all the necessity, perfect-essential necessity, etc) in a different thread on your original comment, if you’re curious!
If you’d like to stop back-and-forthing about metaphysics, seems reasonable. I’m sure we’ll make lots of progress if we keep going debating our priors about this! \s
Yeah I think it might be more productive, if I wanted to make progress on this, to look at the math rather than the metaphysical back-and-forth.
Indeed!