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 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!