It’s either using a different language that happens to use the same alphabet, like if I feed the same string of letters into brains that speak English vs. German (most such strings will be gibberish/errors modulo any given language, of course, but that is also a kind of output which can help us learn the language’s rules and structure), or else it is using the same keys to encode information differently, like typing on a keyboard whose keys are printed as QWERTY layout but which the computer is interpreting as Dvorak layout.
The overwhelming majority of all possible broken calculators are not doing either of those things.
For example, you could have a device where no matter what buttons you push, it always outputs “7”. That is not a substitution cipher on standard arithmetic or a new language; it’s not secretly doing correct math if only you understood how to interpret it. You can’t use it to replace your calculator once you’re trained on it, the way you could start speaking German instead of English, or start typing on a Dvorak keyboard instead of a Qwerty one.
Sorry, I guess those examples weren’t as good as I initially thought. You’re absolutely right. I don’t think that changes the underlying point that no matter what, there exists a correct answer as to what the system is doing, and what it will do in response to any possible input (not necessarily deterministic). But regardless, calling the output “true” or “false” is only relative to the model in your mind of what the inputs and outputs “should” “normally” mean. What is “real” is that the system will behave a certain way, and that you have certain expectations about that behavior which may or may not be accurate, both of which are physical facts as well as logical ones.
I guess I’m also thrown by the OP’s comment that “It doesn’t seem like a primate nervous system comes out of the box with a notion of “reality”.” (Frankly I’d be surprised if we had this as an explicit conscious thought from birth, since it doesn’t seem like the kind of thing evolution would/could have been likely to build in over the short amount of evolutionary time conscious thought has been a thing, or the kind of thing that would be easy to code in DNA). Instead, though, we actually have some pretty strong clues and data as to how/when/whether we and some other species of animals acquire pieces of such notions. Do they recognize themselves in a mirror? If they’re by a mirror and you talk to them, do they turn around or look at your reflection? Do they have object permanence? Do they rely on memory or their senses to find things they want? Can they attempt deception based on information they know but think you or another animal does not? Can they learn signs and/or symbols and use them to share information? Can they do to reference people and things that are not present? How variable are these capabilities within a species?
Then is an expectation that 2 +2 = 3 just as valid as an expectation that 2 + 2 = 4? If so, what is the difference between those statements that makes one of them pragmatically more useful than the other?
No, but an expectation that device A will return “3” in response to the string “2+2″ can be that valid iff device A actually does return “3.”
The thing that makes the 2+2=4 device more useful is that it encodes the same functions using thd same symbols our brains do, functions which we have because we developed formal systems we call arithmetic that correspond to aspects of the world in which we operate. If it were the case that 2+2=3, we’d have developed different formal systems and would build different devices based on them.
that correspond to aspects of the world in which we operate. If it were the case that 2+2=3, we’d have developed different formal systems and would build different devices based on them.
Which aspects are those? What parts of the world could have been different to make 2+2=3 work better than 2+2=4?
You’ve replied 3 times and it seems to me that you have not yet given a clear answer to the original question of where mathematical truth comes from.
It seems obvious to me that 2+2=4 is special in a way that is not contingent on humans. There are no aliens that just happen to use 2+2=3 instead of 2+2=4 and end up with equally good math. So all this talk about correspondence with stuff in human brains seems to me like a distraction.
You’re right, it doesn’t matter that we have human brains. It matters that we evolved in a world where 2+2=3, regardless of humans. But if it did, then it would mean something deep within the structure of the world was different in some way I find hard to imagine, and then I’d have a different kind of mind that evolved in such a different world. And while that specific scenario, 2+2=3, is very hard for me to imagine, it is very *easy for me to imagine scenarios where it could be meaningful to say that 2+2=1 (modular arithmetic, mod 3 in this case) or 1+1=1 (entities that lack discrete structure but instead just merge when combined, like counting clouds before and after they collide) or 2+2>>4 (two pairs of particles with complex interactions that result in a multitude of Everett branches).
My current understanding is that mathematical truths come from the internal relations of the rules of formal systems. Some of those systems are useful because their structure corresponds to aspects of the world, and those are the systems we bother to deeply investigate, give names to, and so on. One of the long-standing open questions in physics is where the laws of physics come from. Einstein: “What really interests me is whether God could have created the world any differently.” I don’t have all that much insight into that beyond the fact that some models of physics try to derive the apparent laws of reality from deeper structures, like the geometry of spacetime, but that just pushes the question back a step.
The overwhelming majority of all possible broken calculators are not doing either of those things.
For example, you could have a device where no matter what buttons you push, it always outputs “7”. That is not a substitution cipher on standard arithmetic or a new language; it’s not secretly doing correct math if only you understood how to interpret it. You can’t use it to replace your calculator once you’re trained on it, the way you could start speaking German instead of English, or start typing on a Dvorak keyboard instead of a Qwerty one.
Sorry, I guess those examples weren’t as good as I initially thought. You’re absolutely right. I don’t think that changes the underlying point that no matter what, there exists a correct answer as to what the system is doing, and what it will do in response to any possible input (not necessarily deterministic). But regardless, calling the output “true” or “false” is only relative to the model in your mind of what the inputs and outputs “should” “normally” mean. What is “real” is that the system will behave a certain way, and that you have certain expectations about that behavior which may or may not be accurate, both of which are physical facts as well as logical ones.
I guess I’m also thrown by the OP’s comment that “It doesn’t seem like a primate nervous system comes out of the box with a notion of “reality”.” (Frankly I’d be surprised if we had this as an explicit conscious thought from birth, since it doesn’t seem like the kind of thing evolution would/could have been likely to build in over the short amount of evolutionary time conscious thought has been a thing, or the kind of thing that would be easy to code in DNA). Instead, though, we actually have some pretty strong clues and data as to how/when/whether we and some other species of animals acquire pieces of such notions. Do they recognize themselves in a mirror? If they’re by a mirror and you talk to them, do they turn around or look at your reflection? Do they have object permanence? Do they rely on memory or their senses to find things they want? Can they attempt deception based on information they know but think you or another animal does not? Can they learn signs and/or symbols and use them to share information? Can they do to reference people and things that are not present? How variable are these capabilities within a species?
Then is an expectation that 2 +2 = 3 just as valid as an expectation that 2 + 2 = 4? If so, what is the difference between those statements that makes one of them pragmatically more useful than the other?
No, but an expectation that device A will return “3” in response to the string “2+2″ can be that valid iff device A actually does return “3.”
The thing that makes the 2+2=4 device more useful is that it encodes the same functions using thd same symbols our brains do, functions which we have because we developed formal systems we call arithmetic that correspond to aspects of the world in which we operate. If it were the case that 2+2=3, we’d have developed different formal systems and would build different devices based on them.
Which aspects are those? What parts of the world could have been different to make 2+2=3 work better than 2+2=4?
You’ve replied 3 times and it seems to me that you have not yet given a clear answer to the original question of where mathematical truth comes from.
It seems obvious to me that 2+2=4 is special in a way that is not contingent on humans. There are no aliens that just happen to use 2+2=3 instead of 2+2=4 and end up with equally good math. So all this talk about correspondence with stuff in human brains seems to me like a distraction.
You’re right, it doesn’t matter that we have human brains. It matters that we evolved in a world where 2+2=3, regardless of humans. But if it did, then it would mean something deep within the structure of the world was different in some way I find hard to imagine, and then I’d have a different kind of mind that evolved in such a different world. And while that specific scenario, 2+2=3, is very hard for me to imagine, it is very *easy for me to imagine scenarios where it could be meaningful to say that 2+2=1 (modular arithmetic, mod 3 in this case) or 1+1=1 (entities that lack discrete structure but instead just merge when combined, like counting clouds before and after they collide) or 2+2>>4 (two pairs of particles with complex interactions that result in a multitude of Everett branches).
My current understanding is that mathematical truths come from the internal relations of the rules of formal systems. Some of those systems are useful because their structure corresponds to aspects of the world, and those are the systems we bother to deeply investigate, give names to, and so on. One of the long-standing open questions in physics is where the laws of physics come from. Einstein: “What really interests me is whether God could have created the world any differently.” I don’t have all that much insight into that beyond the fact that some models of physics try to derive the apparent laws of reality from deeper structures, like the geometry of spacetime, but that just pushes the question back a step.