Suppose we create a second device that looks like a calculator but displays different answers when you push the same buttons. Both devices are equally physical, and you can explain using physics how either of them works. But our common intuitive notion of truth would like to be able to say that one of those devices is giving true answers and the other is giving false answers.
(Or, more rigorously, that one of the devices is completing strings of symbols in a way that conforms to our axioms of arithmetic.)
It’s not clear to me how physics gets you closer to that.
I’ve already linked the sequence in another comment, but Eliezer’s account of the truth of logical statements is given in Logical Pinpointing, and I think I pretty much agree with him.
I don’t think the second device is broken or false. 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. Colloquially, someone who thinks they’re talking to an English speaker or thinks they’re typing on a QWERTY keyboard may find this extremely confusing and think the resulting outputs are false, but they are making an error based on their own limited understanding of the situation and context.
The device may be giving answers that are wrong according to your mental model of what certain symbols mean (aka, your model of arithmetic). But, there exists a mapping from key-presses to outputs, defined by the physical structure of the device itself, that is logical and consistent. In every individual case, there is a well-defined answer to “What will the device output, given a set of key strokes?”, the determination of that output is purely physical, and the way your mind tries to map that relation into a logical understanding or truth value is also purely physical. The same applies to patterns of symbols in a text that your mind receives as input, like reading a math paper containing a proof or a book containing an argument.
The role physics serves, here, is that it’s the part of the logical structure that’s common to every interpreter, every device, and every source of data. It can’t be changed. (Our understanding of it can be changed and refined in response to new data which our previous model of physics failed to correctly predict, of course.) The better you know that part, that base layer, the better you will be at analyzing and understanding the operation of all the other parts.
I would also point out that thermodynamics predates information theory by more than half a century, and has much of the same logical structure for some of the same reasons, and as I understand it von Neumann spotted the similarity immediately when Shannon proposed it even if Shannon wasn’t aware of it. My take is that entropy is in some sense logically prior to any specific set of laws of physics, but that for any set of laws of physics that has enough logical structure to support a universe with life, there will be some equivalent structure that looks like thermodynamics/statistical mechanics/entropy.
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.
People make more working calculators than broken ones. Because they are useful because their answers model other parts of reality. And the axioms are in the human brains. Yes, you can’t derive axioms from physical equations without bridge laws. But you also need bridge laws for anything else high level like utility or physical description of gaining knowledge from experiments. And that is also fine for logic, because you couldn’t perfectly access logical truth by any other procedure anyway, because your implementation of it is failable.
Sure it has—you can both derive from physical equations how calculator works and how notion of contradiction works in human brains.
Suppose we create a second device that looks like a calculator but displays different answers when you push the same buttons. Both devices are equally physical, and you can explain using physics how either of them works. But our common intuitive notion of truth would like to be able to say that one of those devices is giving true answers and the other is giving false answers.
(Or, more rigorously, that one of the devices is completing strings of symbols in a way that conforms to our axioms of arithmetic.)
It’s not clear to me how physics gets you closer to that.
I’ve already linked the sequence in another comment, but Eliezer’s account of the truth of logical statements is given in Logical Pinpointing, and I think I pretty much agree with him.
I don’t think the second device is broken or false. 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. Colloquially, someone who thinks they’re talking to an English speaker or thinks they’re typing on a QWERTY keyboard may find this extremely confusing and think the resulting outputs are false, but they are making an error based on their own limited understanding of the situation and context.
The device may be giving answers that are wrong according to your mental model of what certain symbols mean (aka, your model of arithmetic). But, there exists a mapping from key-presses to outputs, defined by the physical structure of the device itself, that is logical and consistent. In every individual case, there is a well-defined answer to “What will the device output, given a set of key strokes?”, the determination of that output is purely physical, and the way your mind tries to map that relation into a logical understanding or truth value is also purely physical. The same applies to patterns of symbols in a text that your mind receives as input, like reading a math paper containing a proof or a book containing an argument.
The role physics serves, here, is that it’s the part of the logical structure that’s common to every interpreter, every device, and every source of data. It can’t be changed. (Our understanding of it can be changed and refined in response to new data which our previous model of physics failed to correctly predict, of course.) The better you know that part, that base layer, the better you will be at analyzing and understanding the operation of all the other parts.
I would also point out that thermodynamics predates information theory by more than half a century, and has much of the same logical structure for some of the same reasons, and as I understand it von Neumann spotted the similarity immediately when Shannon proposed it even if Shannon wasn’t aware of it. My take is that entropy is in some sense logically prior to any specific set of laws of physics, but that for any set of laws of physics that has enough logical structure to support a universe with life, there will be some equivalent structure that looks like thermodynamics/statistical mechanics/entropy.
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.
People make more working calculators than broken ones. Because they are useful because their answers model other parts of reality. And the axioms are in the human brains. Yes, you can’t derive axioms from physical equations without bridge laws. But you also need bridge laws for anything else high level like utility or physical description of gaining knowledge from experiments. And that is also fine for logic, because you couldn’t perfectly access logical truth by any other procedure anyway, because your implementation of it is failable.
Broken calculators and irrational brains also follow the laws of physics. You would have to know what is normatively correct before you can do that.