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.
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.