That 2+2=4 is a fact about a mathematical system that exists independently of the physical universe, including us humans that decided to use those symbols to express that fact. That fact is in the territory. But, in order to interact with the physical universe, it has to be discovered by some physical system that explores logical conclusions, such as our brains. This exploration builds our map of the territory. Our uncertainty about the tautological statement does not reflect some vagueness in the territory of logic, but our uncertainty about the workings of our physical brains, and their ability to build maps that reflect the territory.
Problems of logic have 100% correct answers, but our physical brains cannot become 100% entangled with those correct answers. It is observation, which can be abstract observations of our own logical reasoning, which give us increasing entanglement which approaches, but never reaches, 100%.
Whatever you could possibly know and value about reality can only exist independently of the physical universe. (Huh?) If your uncertainty about math doesn’t indicate uncertainty of the math, and it’s an argument for math being otherworldly, it’s also an argument for the territory being otherworldly, which is clearly a confusion of terms.
And so you should bring the math back where it belongs, an aspect of the territory.
Whatever you could possibly know and value about reality can only exist independently of the physical universe.
That is not what I am saying. I mean that things that we think of as tautologies, or purely logical truths, which are true no matter what universe we are in, exist independently of the physical universe. Facts about the physical universe are not in this class. Indeed, the entanglement of our physical brains with these logical truths is an example of a fact about the physical universe that, of course, depends on the the universe.
If your uncertainty about math doesn’t indicate uncertainty of the math, and it’s an argument for math being otherworldly...
You have my argument backwards. I first make the point that facts about math are not facts about the physical universe to support that the uncertainty we have about math, which exists in our heads, in our physical universe, does not exist in math itself. The argument does not work the other way, there are plenty of instances of uncertainty in our minds that are not uncertainty in the things elsewhere in the physical universe that they are about.
My comment was an attempt to explain why we need observation to believe things that are objectively true regardless of the world we exists in. Basically, we need evidence that our brains, existing in the physical worlds, are suitable for representing the logical truths.
This is really helpful and I think I agree with all of it. I’ve just never understood “observation” to include my logical reasoning. If your position is that we know 2+2=4 by virtue of observing our own reasoning and not by virtue of any sensory data (information about the outside world) then I don’t think that position is any different from the one I already hold. But is this Eliezer’s position? His OB post made it sound like he could be swayed to think 2+2=3 as a result of external events mediated by his sensory perception of those events. That is what I objected to.
Well, I think that observations can be both our reasoning and sensory data.
Suppose you have a model* of your own accuracy at addition of integers, which is that you are 95% likely to get the correct answer, 2% to be one high, 2% to be one low, and with the remaining 1% divided somehow amongst other possibilities. Then, when you actually observe that when adding 2 + 2 you get 4, this is Bayesian evidence that gives a likelihood ratio of 42.5 : 1 in favor of the theory that 2 + 2 = 4 compared to the theory that 2 + 2 = 3.
Now suppose you have a collection of pebbles, and your model of the pebbles claims that if you count out 2 distinct collections of pebbles, and then combine them and count the total, that the sum of the counts of the distinct collections is 90% likely to be the count of the combined collection, and is 4% likely to be one high, 4% to be one low, and 2% to be something else. And then you actually count out a collection of 2 pebbles, and another collection of 2 pebbles, and combine them, and when you count the combined collection you count 4 pebbles. This is Bayesian evidence with a likelihood ratio of 22.5 : 1 in favor of 2 + 2 = 4 as opposed to 2 + 2 = 3.
In both cases, belief in a logical proposition results from our belief that an observable system has some probability of reflecting logical truth. If, as in the example numbers that I made up just now, we believe that our reasoning process is more likely than observations of our environment, then the results of our reasoning is stronger evidence, but it is still the same class of evidence.
* I have neglected the harder problem of simultaneously updating propositions about additions and propositions about a given system’s probability of representing addition. That is, I have not explained where the models I asked you suppose you have really should come from.
That 2+2=4 is a fact about a mathematical system that exists independently of the physical universe, including us humans that decided to use those symbols to express that fact. That fact is in the territory. But, in order to interact with the physical universe, it has to be discovered by some physical system that explores logical conclusions, such as our brains. This exploration builds our map of the territory. Our uncertainty about the tautological statement does not reflect some vagueness in the territory of logic, but our uncertainty about the workings of our physical brains, and their ability to build maps that reflect the territory.
Problems of logic have 100% correct answers, but our physical brains cannot become 100% entangled with those correct answers. It is observation, which can be abstract observations of our own logical reasoning, which give us increasing entanglement which approaches, but never reaches, 100%.
Whatever you could possibly know and value about reality can only exist independently of the physical universe. (Huh?) If your uncertainty about math doesn’t indicate uncertainty of the math, and it’s an argument for math being otherworldly, it’s also an argument for the territory being otherworldly, which is clearly a confusion of terms.
And so you should bring the math back where it belongs, an aspect of the territory.
That is not what I am saying. I mean that things that we think of as tautologies, or purely logical truths, which are true no matter what universe we are in, exist independently of the physical universe. Facts about the physical universe are not in this class. Indeed, the entanglement of our physical brains with these logical truths is an example of a fact about the physical universe that, of course, depends on the the universe.
You have my argument backwards. I first make the point that facts about math are not facts about the physical universe to support that the uncertainty we have about math, which exists in our heads, in our physical universe, does not exist in math itself. The argument does not work the other way, there are plenty of instances of uncertainty in our minds that are not uncertainty in the things elsewhere in the physical universe that they are about.
My comment was an attempt to explain why we need observation to believe things that are objectively true regardless of the world we exists in. Basically, we need evidence that our brains, existing in the physical worlds, are suitable for representing the logical truths.
This is really helpful and I think I agree with all of it. I’ve just never understood “observation” to include my logical reasoning. If your position is that we know 2+2=4 by virtue of observing our own reasoning and not by virtue of any sensory data (information about the outside world) then I don’t think that position is any different from the one I already hold. But is this Eliezer’s position? His OB post made it sound like he could be swayed to think 2+2=3 as a result of external events mediated by his sensory perception of those events. That is what I objected to.
Well, I think that observations can be both our reasoning and sensory data.
Suppose you have a model* of your own accuracy at addition of integers, which is that you are 95% likely to get the correct answer, 2% to be one high, 2% to be one low, and with the remaining 1% divided somehow amongst other possibilities. Then, when you actually observe that when adding 2 + 2 you get 4, this is Bayesian evidence that gives a likelihood ratio of 42.5 : 1 in favor of the theory that 2 + 2 = 4 compared to the theory that 2 + 2 = 3.
Now suppose you have a collection of pebbles, and your model of the pebbles claims that if you count out 2 distinct collections of pebbles, and then combine them and count the total, that the sum of the counts of the distinct collections is 90% likely to be the count of the combined collection, and is 4% likely to be one high, 4% to be one low, and 2% to be something else. And then you actually count out a collection of 2 pebbles, and another collection of 2 pebbles, and combine them, and when you count the combined collection you count 4 pebbles. This is Bayesian evidence with a likelihood ratio of 22.5 : 1 in favor of 2 + 2 = 4 as opposed to 2 + 2 = 3.
In both cases, belief in a logical proposition results from our belief that an observable system has some probability of reflecting logical truth. If, as in the example numbers that I made up just now, we believe that our reasoning process is more likely than observations of our environment, then the results of our reasoning is stronger evidence, but it is still the same class of evidence.
* I have neglected the harder problem of simultaneously updating propositions about additions and propositions about a given system’s probability of representing addition. That is, I have not explained where the models I asked you suppose you have really should come from.