Axiomatic Systems … can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I’m not even sure what you mean by it.
However, I think you’re confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
At the very least this warrants further explication.
Axiomatic Systems … can all be reduced to physics. I think most LessWrongers, being reductionists, believe this.
I would be suprised if this were true. In fact, I’m not even sure what you mean by it.
Well, given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical.
However, I think you’re confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
You can specify models much bigger than the physical universe. But that’s just extrapolating the rules by assuming they would keep on working. We do have good reason to believe that they would keep on working, though, because if they stop, then contradictions take place, and from contradictions anything is true, so we would be living in one very strange universe.
Edit:
It has also occurred to me that it doesn’t even matter if a number is larger than the total amount of atoms in this universe. Because as I’ve said in my post, a number is what you get when you abstract away all the little details which aren’t shared by all the places where numbers emerge, like the fact that it’s atoms you’re counting.
So a system for representing a number larger than the total number of atoms represents a number anyway, as long as it follows the rules of numbers. And a model of something much bigger than the universe works simply because the details of how large the universe actually is are ignored in the model.
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
Indeed, the mind seems empirically to be something we experience due to the workings of visibly finite machine.
My mind contains a concept of 4 which is pretty dang useful, I can visualize 4 in so many ways and see it so often in reality without even trying to look for it. My mind’s concept of 1000 doesn’t suck, but is starting to look more like fluid measure (continuously variable) than discrete. With neocortical help I can improve my concept of 1000 around the edges, learning what 1000 dollars looks like, or a thousand grains of sand, or 1000 people in a stadium, but the amount of the conception of 1000 that my mind actually misses, when held side by side with my conception of 4 is, I think, very obvious.
I can think of 10 billion, which is more than the neurons in my brain, 100 billion which is more than the cells in my body (including the bacteria), 10 trillion (which is more than the neuronal interconnections in my brain) but all of these concepts are so fuzzy as to be honored primarily in the logarithm base 10. That is, virtually the only thing I know about 10 billion is that it is 10x as much as 1 billion, and similar such indirect conceptions.
My point in this is that if we think about how we actually think about numbers, the physicalism seems clear, including the limits in clarity we might expect as we exceed the number of parts of the physical system we are using for doing such thinking.
So how, then, do we come up with theorems about different kinds of infinities if our conceptions of these things are so finite? I believe it to be true (please correct me) that
1) the number of things we know about infinities is quite finite (maybe 1000 things at most?)
2) the proofs we use to know these finite number of things about infinities are also quite finite (comprising perhaps 10,000 pages, perhaps 100,000 to prove everything humans know about math, including everything we know about infinities)
PUNCHLINE:
Therefore it is very suggestive to think the physical substrate is a gigantic part of the story, and I am at a loss to see an opening for any serious contribution from a non-physical source.
ETA: minds can contain gods, magic and any number of wonders that are fundamentally irreconcilable with physical reality
Are not the non-existence of gods and magic empirical truths? I can imagine someone with a map tailored to the universe-as-is would say if they started seeing magic and gods that there was a deeper sense in which the magic was really technology we don’t understand and the gods were life that exceeded our abilities in dimensions where the individual from our world had never seen such strong evidence of any life even tying humans in ability.
But I should think in a universe with beings who exceeded our intelligence by factors of a trillion that magic and gods would be a lot better map of that territory than the map we use in our universe without such beings. Is a dog who believes “slavery is wrong” actually better in some real way in a world like ours (where dogs are not very smart and are domesticated by humans)?
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
...
ETA: minds can contain gods, …
No, I’m claiming that the idea of god exists physically.
In our universe, the map is part of the territory. So the concept of god which a human stores in his mind is something physical. God himself might not exist, but the idea of god, and the rules this idea follows, exist, despite being inconsistent. And these rules which the idea of god follows can be represented in many ways, all of them physical.
For example, in the human mind, in computers, in mathematical logic (despite inconsistencies), etc. All these ways of representing god are done using completely different configurations of molecules. What is the common ground between them? Certainly not that the idea of god and it’s rules are some special thing with special properties. So what do the hard drive and the human mind have in common when representing the idea of god?
By my theory, what they have in common is abstraction. Ignore all the specific details about how hard drives and human minds work, and just look at the specific abstract rules which we define as “god”. These are complex, so we can’t easily visualize this removal of details. It’s much easier when talking about apples and numbers. You can see that when you have 2 apples, you can get the idea of 2 by ignoring the fact that it’s apples, and that they’re in a bag, and that gravity is affecting them. It’s also easy to see when talking about balls. You get the idea of a ball by taking a sphere of matter, forgetting what it’s composed of and forgetting it’s radius. This abstract idea of a “ball” fits many things, because it’s just ignoring details which vary from ball to ball.
So my claim is that the idea of axiomatic systems exists in the physical universe. In fact, all the ideas we ever have, and there rules, exist in the physical universe. But if we take PA as an example, the idea of PA exists in a mathematician’s mind, and numbers emerge inside this idea of PA, because numbers do emerge inside PA. So by removing the details of how PA is stored in the mathematician’s mind, we obtain numbers, which is just like getting numbers by removing the details about apples.
This still leaves the question of why numbers emerge in so many places. My best guess is that they do because the universe is built upon simple and universal laws of physics, so it’s only natural that the same patterns would be appear everywhere.
So a system for representing a number larger than the total number of atoms represents a number anyway,
There seems to be no problem representing numbers using machines that have many fewer pieces than the number represented. With only 10 bits I can represent more than 1000 numbers as a trivial example. WIth only 10 billion neurons I can represent infinity (in the human mind) although it might be difficult to prove my representation was perfectly accurate.
I would be suprised if this were true. In fact, I’m not even sure what you mean by it.
However, I think you’re confusing the finitude of our proofs with some sort of property of the models. I mean, I can easily specify models much bigger than the physical universe.
At the very least this warrants further explication.
Well, given that mathematicians store axiomatic systems in their minds, and use them to prove things, they cannot help but be reducible to physical things, unless the mind itself is not physical.
You can specify models much bigger than the physical universe. But that’s just extrapolating the rules by assuming they would keep on working. We do have good reason to believe that they would keep on working, though, because if they stop, then contradictions take place, and from contradictions anything is true, so we would be living in one very strange universe.
Edit:
It has also occurred to me that it doesn’t even matter if a number is larger than the total amount of atoms in this universe. Because as I’ve said in my post, a number is what you get when you abstract away all the little details which aren’t shared by all the places where numbers emerge, like the fact that it’s atoms you’re counting.
So a system for representing a number larger than the total number of atoms represents a number anyway, as long as it follows the rules of numbers. And a model of something much bigger than the universe works simply because the details of how large the universe actually is are ignored in the model.
Is your claim that because the mind is itself physical, any idea stored in a mind is necessarily reducible to something physical?
Because this seems like a map-territory confusion.
ETA: minds can contain gods, magic and any number of wonders that are fundamentally irreconcilable with physical reality.
Indeed, the mind seems empirically to be something we experience due to the workings of visibly finite machine.
My mind contains a concept of 4 which is pretty dang useful, I can visualize 4 in so many ways and see it so often in reality without even trying to look for it. My mind’s concept of 1000 doesn’t suck, but is starting to look more like fluid measure (continuously variable) than discrete. With neocortical help I can improve my concept of 1000 around the edges, learning what 1000 dollars looks like, or a thousand grains of sand, or 1000 people in a stadium, but the amount of the conception of 1000 that my mind actually misses, when held side by side with my conception of 4 is, I think, very obvious.
I can think of 10 billion, which is more than the neurons in my brain, 100 billion which is more than the cells in my body (including the bacteria), 10 trillion (which is more than the neuronal interconnections in my brain) but all of these concepts are so fuzzy as to be honored primarily in the logarithm base 10. That is, virtually the only thing I know about 10 billion is that it is 10x as much as 1 billion, and similar such indirect conceptions.
My point in this is that if we think about how we actually think about numbers, the physicalism seems clear, including the limits in clarity we might expect as we exceed the number of parts of the physical system we are using for doing such thinking.
So how, then, do we come up with theorems about different kinds of infinities if our conceptions of these things are so finite? I believe it to be true (please correct me) that 1) the number of things we know about infinities is quite finite (maybe 1000 things at most?) 2) the proofs we use to know these finite number of things about infinities are also quite finite (comprising perhaps 10,000 pages, perhaps 100,000 to prove everything humans know about math, including everything we know about infinities)
PUNCHLINE: Therefore it is very suggestive to think the physical substrate is a gigantic part of the story, and I am at a loss to see an opening for any serious contribution from a non-physical source.
Are not the non-existence of gods and magic empirical truths? I can imagine someone with a map tailored to the universe-as-is would say if they started seeing magic and gods that there was a deeper sense in which the magic was really technology we don’t understand and the gods were life that exceeded our abilities in dimensions where the individual from our world had never seen such strong evidence of any life even tying humans in ability.
But I should think in a universe with beings who exceeded our intelligence by factors of a trillion that magic and gods would be a lot better map of that territory than the map we use in our universe without such beings. Is a dog who believes “slavery is wrong” actually better in some real way in a world like ours (where dogs are not very smart and are domesticated by humans)?
No, I’m claiming that the idea of god exists physically.
In our universe, the map is part of the territory. So the concept of god which a human stores in his mind is something physical. God himself might not exist, but the idea of god, and the rules this idea follows, exist, despite being inconsistent. And these rules which the idea of god follows can be represented in many ways, all of them physical.
For example, in the human mind, in computers, in mathematical logic (despite inconsistencies), etc. All these ways of representing god are done using completely different configurations of molecules. What is the common ground between them? Certainly not that the idea of god and it’s rules are some special thing with special properties. So what do the hard drive and the human mind have in common when representing the idea of god?
By my theory, what they have in common is abstraction. Ignore all the specific details about how hard drives and human minds work, and just look at the specific abstract rules which we define as “god”. These are complex, so we can’t easily visualize this removal of details. It’s much easier when talking about apples and numbers. You can see that when you have 2 apples, you can get the idea of 2 by ignoring the fact that it’s apples, and that they’re in a bag, and that gravity is affecting them. It’s also easy to see when talking about balls. You get the idea of a ball by taking a sphere of matter, forgetting what it’s composed of and forgetting it’s radius. This abstract idea of a “ball” fits many things, because it’s just ignoring details which vary from ball to ball.
So my claim is that the idea of axiomatic systems exists in the physical universe. In fact, all the ideas we ever have, and there rules, exist in the physical universe. But if we take PA as an example, the idea of PA exists in a mathematician’s mind, and numbers emerge inside this idea of PA, because numbers do emerge inside PA. So by removing the details of how PA is stored in the mathematician’s mind, we obtain numbers, which is just like getting numbers by removing the details about apples.
This still leaves the question of why numbers emerge in so many places. My best guess is that they do because the universe is built upon simple and universal laws of physics, so it’s only natural that the same patterns would be appear everywhere.
There seems to be no problem representing numbers using machines that have many fewer pieces than the number represented. With only 10 bits I can represent more than 1000 numbers as a trivial example. WIth only 10 billion neurons I can represent infinity (in the human mind) although it might be difficult to prove my representation was perfectly accurate.