This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise. I’ll try to summarize the context in which the idea of mathematical universe looks to me this way.
When abstract objects or ideas are thought about with mathematical precision, it turns out that they are best described by their “structure”, which is a collection of properties that these things have (like commutativity of multiplication on a complex plane or connectedness of a sphere), rather than some kind of “reductionistic” recipe for assembling them. These properties imply other properties, and in many interesting cases, even based on a fixed initial definition it’s possible to explore them in many possible ways, there is no restriction to a single direction in finding more properties (like new laws of number theory or geometry, as opposed to running a computer program to completion). At the same time, it’s not possible, either in principle or in practice, to infer all interesting properties following from given defining properties that specify a sufficiently complicated structure, so there is perpetual logical uncertainty.
When two structures (or two “things” having these respective structures, described by them to some extent) share some of their properties in some sense, it’s possible to infer new facts about one of the structures by observing the other. This way, for example, a computer program can reason about an infinite structure: if we know that a certain property stands or falls for the program and for the structure together, we can conclude that the property holds for the structure if its counterpart does for the program and so on. Also, setting up a structure that reflects properties of another one doesn’t require knowing all defining properties of that structure, knowing only sufficiently accurate approximations to some of them may be sufficient to make useful inferences.
Physical world then can be seen as just another thing that, to the extent it can be rigorously thought about, is described by certain properties or principles, of which we know only some and not precisely. Thinking about the world involves setting up certain things (brains, computers, experimental apparatus, abstract structures, physical theories, etc.) that capture some of its structure (these act as “maps” of the world), and then inferring more properties (making “predictions”) based on what they’ve managed to capture.
It doesn’t seem like there is much more to say on this big picture level, and treating physical world the same way we treat other complicated things, such as sufficiently complicated mathematical structures, seems like a natural thing to do. Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful. The physical world has structure, just as arithmetic has structure, but it doesn’t seem like much more can be said on this level of description.
Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful.
Every existing thing has a structure, but it is not clear that every logically consistent structure is the structure of an existing thing. The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless. The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
A way to tell an instantiated mathematical-structure-containing-sentient-beings from an uninstantiated one. (That doesn’t sound very different from telling conscious beings from philosophical zombies to me.)
I don’t know whether the concept of existence is meaningful. If it is, then something like the following should work:
To determine whether a mathematical structure M is instantiated, examine every thing that exists. If M is the structure of something that you examine, then M is instantiated. Otherwise, M is not instantiated.
Thus, whether the concept of existence is meaningful is the heart of the problem. I don’t claim to know that this concept is meaningful. I claim only not to know that it is meaningless.
I think it’s more like there are several concepts which share the same label. If a tree falls in the forest, and no one hears it, does it make a sound?
The tree in the forest is a case of various clear concepts (of sound) clearly implying different true answers.
The problem of Being is a problem of finding a clear concept that implies answers that many people find intuitively plausible.
It is more like the problem of being perfectly confident that various mathematical statements are true, while finding it very difficult to say just what it is that those statements are true about.
*points at objects which are instances of a class*
Those are instantiated (classes).
*points at classes that are unused at runtime, do not have any real object instances, perhaps were never even coded, but are simply logically consistent*
The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless.
The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
I’m making a distinction between saying “physical world is a structure” and “physical world has structure”: the first form seems to demand something unclear, and the latter seems to suffice for all purposes. Suppose things may either exist or not; but structure of things is abstract math, so it does seem clear that the properties of a structure don’t care whether it’s “instantiated” or not: the math works out according to what the structure is, regardless of which things have it. And since we only reason about things in terms of their structure, a distinction that isn’t reflected in that structure can’t enter into our reasoning about them.
(It might be possible to cash out “existence” of the kind physical world has as a certain property of structures, probably something very non-fundamental, like human morality, but this interpretation seems unlike the kind of confusion the argument is meant to counter.)
This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise.
I can’t find anything to disagree with after this quoted sentence, but “this seems like a trivial idea” certainly isn’t something I’d say if someone else wrote the comment you’re replying to. My guess is that you think “makes decision theory much easier” gives Tegmark too much credit because decision theory is far from solved, there are lots of hard problems left, and Tegmark’s ideas represent only a small step, in a relative sense, compared to the overall difficulty of the project.
If my guess is right, I could offer the defense that it feels like a large amount of progress to me, in an absolute sense, but it might be a good idea to just rephrase that sentence to avoid giving the wrong impression. Or, let me know if I’m totally off base and you intended a different point entirely.
I mean only that the description I sketched (which might be seen as referring the the idea of “mathematical universe”, but also deconstructs some of it, suggesting that it’s meaningless to insist that something “is a mathematical structure”), isn’t saying much of anything, and uses only standard ideas from mathematics; in this sense the idea of “mathematical universe” doesn’t say much of anything either (i.e. is trivial).
It might be a useful point to the extent that understanding it would banish useless ways of metaphysical theorizing about the physical world and free up time for more fruitful activities. So, my comment is unrelated to your point about decision theory, although the simplification (back to triviality) may be useful there and probably more relevant than for most other problems.
This seems like a trivial idea, interesting mostly insofar as it dispels unnecessary mysteriousness of physical world, but not particularly meaningful or helpful otherwise. I’ll try to summarize the context in which the idea of mathematical universe looks to me this way.
When abstract objects or ideas are thought about with mathematical precision, it turns out that they are best described by their “structure”, which is a collection of properties that these things have (like commutativity of multiplication on a complex plane or connectedness of a sphere), rather than some kind of “reductionistic” recipe for assembling them. These properties imply other properties, and in many interesting cases, even based on a fixed initial definition it’s possible to explore them in many possible ways, there is no restriction to a single direction in finding more properties (like new laws of number theory or geometry, as opposed to running a computer program to completion). At the same time, it’s not possible, either in principle or in practice, to infer all interesting properties following from given defining properties that specify a sufficiently complicated structure, so there is perpetual logical uncertainty.
When two structures (or two “things” having these respective structures, described by them to some extent) share some of their properties in some sense, it’s possible to infer new facts about one of the structures by observing the other. This way, for example, a computer program can reason about an infinite structure: if we know that a certain property stands or falls for the program and for the structure together, we can conclude that the property holds for the structure if its counterpart does for the program and so on. Also, setting up a structure that reflects properties of another one doesn’t require knowing all defining properties of that structure, knowing only sufficiently accurate approximations to some of them may be sufficient to make useful inferences.
Physical world then can be seen as just another thing that, to the extent it can be rigorously thought about, is described by certain properties or principles, of which we know only some and not precisely. Thinking about the world involves setting up certain things (brains, computers, experimental apparatus, abstract structures, physical theories, etc.) that capture some of its structure (these act as “maps” of the world), and then inferring more properties (making “predictions”) based on what they’ve managed to capture.
It doesn’t seem like there is much more to say on this big picture level, and treating physical world the same way we treat other complicated things, such as sufficiently complicated mathematical structures, seems like a natural thing to do. Of course, the physical world is very special, it is this particular thing with these particular properties, and we happen to have evolved and live in it, but that doesn’t seem fundamentally different from how the complex plane is another particular thing with its own properties. Also, like “physical” is not a meaningful distinction in the sense that it doesn’t say anything specific about properties of the world, also “mathematical structure” is not a meaningful distinction in the same sense, and so insisting that the physical world “is a mathematical structure” doesn’t seem meaningful. The physical world has structure, just as arithmetic has structure, but it doesn’t seem like much more can be said on this level of description.
Every existing thing has a structure, but it is not clear that every logically consistent structure is the structure of an existing thing. The distinction between instantiated and uninstantiated mathematical structures is not obviously meaningless. The Tegmark hypothesis is that this distinction is meaningless. Since this meaninglessness is not obvious, the Tegmark hypothesis is nontrivial.
Define instantiated.
What would constitute a definition for your purposes?
A way to tell an instantiated mathematical-structure-containing-sentient-beings from an uninstantiated one. (That doesn’t sound very different from telling conscious beings from philosophical zombies to me.)
I don’t know whether the concept of existence is meaningful. If it is, then something like the following should work:
To determine whether a mathematical structure M is instantiated, examine every thing that exists. If M is the structure of something that you examine, then M is instantiated. Otherwise, M is not instantiated.
Thus, whether the concept of existence is meaningful is the heart of the problem. I don’t claim to know that this concept is meaningful. I claim only not to know that it is meaningless.
I think it’s more like there are several concepts which share the same label. If a tree falls in the forest, and no one hears it, does it make a sound?
The tree in the forest is a case of various clear concepts (of sound) clearly implying different true answers.
The problem of Being is a problem of finding a clear concept that implies answers that many people find intuitively plausible.
It is more like the problem of being perfectly confident that various mathematical statements are true, while finding it very difficult to say just what it is that those statements are true about.
*points at objects which are instances of a class*
Those are instantiated (classes).
*points at classes that are unused at runtime, do not have any real object instances, perhaps were never even coded, but are simply logically consistent*
Those are uninstantiated (classes).
Perhaps that’ll help seeing it.
I’m making a distinction between saying “physical world is a structure” and “physical world has structure”: the first form seems to demand something unclear, and the latter seems to suffice for all purposes. Suppose things may either exist or not; but structure of things is abstract math, so it does seem clear that the properties of a structure don’t care whether it’s “instantiated” or not: the math works out according to what the structure is, regardless of which things have it. And since we only reason about things in terms of their structure, a distinction that isn’t reflected in that structure can’t enter into our reasoning about them.
(It might be possible to cash out “existence” of the kind physical world has as a certain property of structures, probably something very non-fundamental, like human morality, but this interpretation seems unlike the kind of confusion the argument is meant to counter.)
I can’t find anything to disagree with after this quoted sentence, but “this seems like a trivial idea” certainly isn’t something I’d say if someone else wrote the comment you’re replying to. My guess is that you think “makes decision theory much easier” gives Tegmark too much credit because decision theory is far from solved, there are lots of hard problems left, and Tegmark’s ideas represent only a small step, in a relative sense, compared to the overall difficulty of the project.
If my guess is right, I could offer the defense that it feels like a large amount of progress to me, in an absolute sense, but it might be a good idea to just rephrase that sentence to avoid giving the wrong impression. Or, let me know if I’m totally off base and you intended a different point entirely.
I mean only that the description I sketched (which might be seen as referring the the idea of “mathematical universe”, but also deconstructs some of it, suggesting that it’s meaningless to insist that something “is a mathematical structure”), isn’t saying much of anything, and uses only standard ideas from mathematics; in this sense the idea of “mathematical universe” doesn’t say much of anything either (i.e. is trivial).
It might be a useful point to the extent that understanding it would banish useless ways of metaphysical theorizing about the physical world and free up time for more fruitful activities. So, my comment is unrelated to your point about decision theory, although the simplification (back to triviality) may be useful there and probably more relevant than for most other problems.