You’ve erred from the moment you think that that a true statement implies independent existence of its predicates, leaving out the possibility that the predicates exist only as part of models.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”. See my reply to wedrifid.
Isn’t that exactly what I did? The method I gave is that you break down the truth assertion into a claim about the correctness of the terms (i.e., part 1, what the common usage is), and a claim about whether the model in common usage implies the truth (part 2). I suppose I could have been more specific about how exactly one does that, but it seems straightforward enough that I didn’t think I needed to give more detail.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”.
How does “3 is prime” imply that “3″ exists, while “primeness is related to the zeroes of the Zeta function” not imply that “primeness” exists?
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
How does “3 is prime” imply that “3″ exists, while “primeness is membership in the set of zeroes of the Zeta function” not imply that “primeness” exists?
The latter does imply primeness exists. But “3 is prime” doesn’t. Luckily you haven’t just used primeness as the value of a bound variable, you’ve given an appropriate paraphrase (although now you’re committed to the existence of the set of zeros of the Zeta function).
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
My method would be to find the associated #1 and #2 statements. The #1 statement would be a claim about what people use the terms “3”, “is”, and “prime” to mean. Under this method you would next identify the common conception of “3″ (by empirical examination of how people use the term and under constraint of Occam’s Razor) as something like, “the quantity immediately following the quantity immediately following the quantity immediately following the quantity of nothing”. Then do the same for the other parts.
(Also, keep in mind that this method is only necessary for the bare statement that “3 is prime”. You needn’t construct the associated #1 statement for more specific claims like, “Here is a system of math. Under those rules and definitions, 3 is prime.”)
Then you would construct the #2 statement, which would be that, under those meanings, the claim as a whole follows from the definitions and assumptions of the system implictly used by those meanings. This would be something like, “under any physical system behaving isomorphically to the assumptions in #1, the physical correlate of ‘3 being prime’ will hold”, and that physical correlate will be something like, “any division of the units correlating to 3 will be such that each partition will have a different number of units, or one unit, or three units”.
...Er, okay, perhaps more detail was needed. Does that answer your question, though?
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.
No, no. Not the predicates. Just the values of it’s bound variables. I’m not saying “Primeness exists”. See my reply to wedrifid.
So “3 is prime” means what? “2+3= 5″ means what? You absolutely need to give more detail because as far as I know none of the many very smart people who have tried have been able to give a satisfactory paraphrase for all true mathematical statements.
How does “3 is prime” imply that “3″ exists, while “primeness is related to the zeroes of the Zeta function” not imply that “primeness” exists?
This whole discussion is, ultimately, about the definition of the word “exist”. But if you try to hang that definition off linguistic phenomena, then you’re at the whim of every linguistic construct people can come up with, and people can probably twist words to get something covered by your definition that you didn’t want to be.
The latter does imply primeness exists. But “3 is prime” doesn’t. Luckily you haven’t just used primeness as the value of a bound variable, you’ve given an appropriate paraphrase (although now you’re committed to the existence of the set of zeros of the Zeta function).
Huh?
My method would be to find the associated #1 and #2 statements. The #1 statement would be a claim about what people use the terms “3”, “is”, and “prime” to mean. Under this method you would next identify the common conception of “3″ (by empirical examination of how people use the term and under constraint of Occam’s Razor) as something like, “the quantity immediately following the quantity immediately following the quantity immediately following the quantity of nothing”. Then do the same for the other parts.
(Also, keep in mind that this method is only necessary for the bare statement that “3 is prime”. You needn’t construct the associated #1 statement for more specific claims like, “Here is a system of math. Under those rules and definitions, 3 is prime.”)
Then you would construct the #2 statement, which would be that, under those meanings, the claim as a whole follows from the definitions and assumptions of the system implictly used by those meanings. This would be something like, “under any physical system behaving isomorphically to the assumptions in #1, the physical correlate of ‘3 being prime’ will hold”, and that physical correlate will be something like, “any division of the units correlating to 3 will be such that each partition will have a different number of units, or one unit, or three units”.
...Er, okay, perhaps more detail was needed. Does that answer your question, though?
Yes. That is helpful. I was going to bring this up in my first comment but decided to focus on one thing at a time. Your view seems very similar to nominalist-structuralism, which I find appealing as well. At least I can read nominalist-structuralism into your post and comments. Afaict, it’s considered the most promising version of nominalism going. They basically take your use of isomorphism and go one step farther. The SEP discusses it some but it’s a pretty poorly written article. You might have to do a lot of googling. Structuralism argues that math does not describe objects of any kind but rather, structures and places within structures (and have no identity or features outside those structures). Any given integer is not an object but a places in the structure of integers. As you can imagine once you think of mathematical truths this way it become obvious how math can be used to describe other physical systems: namely those systems instantiate the same structure (or in your words, are isomorphic?). Nominalist structuralism involves disclaiming the existence of structures (which are abstract objects) as independent of the systems that instantiate them
ETA: One issue here is the work being done by the word “resembles” when we say “the structure of real numbers resembles the structure of space’, or “the structure of macroscopic object motion resembles the structure of simulated object motion in your physics simulator”. Which is the same issue as “what is the work being done by ‘isomorphic’.