I have two different answers to your question: one practical, one more theoretical. On a practical level, what I gain from peak experiences depends on where my attention is. If I’m out and about, or doing something materially, then the main advantage I gain is noticing new aspects of a situation, or seeing the same aspects in a different light; I believe this is a result of greater flexibility in choosing the cognitive map I apply to the territory. These, I suppose, would be the “unknown dots”: information that was present in the environment, but which my brain never bothered to record. If I’m sitting around thinking, on the other hand, then I tend to find a lot of unconventional connections. Even here, though, there is new information to be gained; I’ve learned a lot about my own nervous system simply by careful observation of my internal experience.
On a theoretical level, I have some trouble with the distinction between information and deduction. In the strictest sense, mathematical truths contain zero information, since they are automatically true in every possible world (or insert your own X-Rationalist translation of this claim). Yet we are still surprised to learn, for example, that e^(pi.i)+1=0 - or I know I was surprised, at the very least. I think this is a result of the fact that our evaluation of mathematical claims is based on manipulation of tangible stuff: we can check our memory to determine if we’ve ever seen a proof before, or we can manipulate symbolic expressions, encoded in our wetware, to attempt to prove or disprove it. Even wood pulp and graphite can be leveraged for this purpose, and so the result of this computation comes in as an observation of an uncertain outcome in the world.
Yet the no-information claim takes on new complications when we realize that our brains use the same basic processes to encode relations between facts, as it does to encode facts. This leads to kind of “flat” ontology, in which we can treat relations as facts and draw relations between them, or even between a relation and a fact. We can even draw relations between internal experience (including mathematical knowledge) and external information. Once we have recognized these connections, natural systems can actually point us toward mathematical truths (for example: we might never have learned about solitons if we had not observed them in nature). The hierarchy of “levels of organization”, and the division between information and deduction from information, therefore appear to be constructed post hoc, and I’m not sure if I can cleanly distinguish between unknown dots and unknown relations.
I’ll attempt a translation. If I’m engaging with the world, then I notice new things about it, or I see things in new ways. For example: once, looking at the sky, I noticed that it was brightest near the horizon and darkest at the zenith. Suddenly I realized the reason: there was more air between me and the horizon than there was between me and the space directly above me. The scene snapped into focus, and I found I could distinctly see the atmosphere as a three-dimensional mass. If I turn my attention inward, on the other hand, I tend to draw connections between pieces of information I already have—suddenly intuiting the behaviour of quantum wave packets, for example, or drawing an analogy between my social networking behaviour and annealing. These two cases are the “dots” and the connections between the dots, respectively.
Information theory generally defines information to be about some event with an uncertain outcome: if you know a coin has been flipped, you will need additional information to determine whether it came up heads or tails. By contrast, if you already know that five coins were flipped and three came up heads, you don’t need any additional information to deduce that the number of heads was prime. Anything that you could in principle figure out from the information you’ve already got isn’t treated as new information; In this sense, mathematical truths (connections) are separated from information proper (“dots”).
While this separation may be useful for theory, it doesn’t capture all aspects of the way we learn and process information. For starters, we’re often rather surprised to learn mathematical facts; this is because we need to use physical hardware (our brains) to compute proofs, and we don’t know what the outcome of the computation will be. Also, our brains seem to treat things, states, patterns, and pieces of information all in the same way—hence, for example, we can refer to “the economy” as if it were a single thing rather than a complex system of interrelationships; or, going in the opposite direction, we can break down a tree into its component cells and, moreover, recognize each cell as a fantastically complicated system, and so on.
Meanwhile, our ability to draw links between different levels of organization allows us, in particular, to see that certain mathematical patterns are reflected the world around us. Once we’ve found a model that fits the system, we can make predictions we couldn’t make before: the more confident I am, say, that every fifth coin flip will come up as heads, the less information will be conveyed when this does indeed happen. It goes in the other direction too: sometimes we see patterns in nature which point us toward new mathematical understanding. The example I gave was of soliton waves, which you can read about here—even if you have no technical background, I think you’ll find the History section enlightening.
For all these reasons, I suspect that a better model of information might loosen the hard distinction that’s made between new information and new deductions.
mathematical truths contain zero information, since they are automatically true in every possible world
This is a false assertion, they are only true if the axioms used to conclude them correspond to reality. There are proofs that rely on the Axiom of Choice which is not accepted (as far as I can tell) by everyone on this site (as well as the axiom of infinity?). There are proofs that rely on the GCH or Large Cardinal Axioms or V=L which are not among the accepted axioms and proven to be independent of the other axioms.
This is a fair point, but I’m referring to information in the information theoretic sense; in this technical sense, mathematical truths are indeed not information.
There are proofs that rely on the GCH or Large Cardinal Axioms or V=L which are not among the accepted axioms and proven to be independent of the other axioms.
I’m aware that the Axiom of Choice is required for some important results of practical import (Tychonoff’s theorem, for example, is equivalent to it), but do you know of any important and useful results following from the GCH, etc.? I’ve only looked into this a little; foundational math is not really my field.
Game theoretic results that are generalized to infinite games often require the use of the GCH. For instance see “Variations on a Game” by J Beck 1981.
It seems Skatche is roughly talking about what is called semantic entailment in logic and I would partially agree with your criticism… since mathematical truth is considered to be more than just that, it includes axioms that you feel good about accepting. However, I’m not sure where reality comes into the picture when considering the definition of mathematical truth.
How do we choose axioms that are good to accept as valid except for our experience with reality? This is a philosophical question so isn’t often considered within mathematics. If one reads some of Eliezer’s posts where he doubts the axiom of infinity (unless I am misunderstanding what he is doing) then it becomes clear that his argument for doubting the axiom is that it doesn’t correspond with reality.
This is generally why choice is doubted; as it is thought that non-measurable sets do not play any role in reality they are ignored in standard statistics and calculus. If non-measurable sets do play a role within the real world then some very odd things can happen that so far no one has observed happen.
I suppose if mathematical truth is changed to be whatever is provable given all possible combinations of non-contradictory axioms then reality does not play any role in math.
I think when trying to study information and its relation to mathematical truth, we must start off practical and should be talking about provability in formal systems of logic. I don’t actually know of any rigorous connections between the two notions, but I can think of an argument that “mathematical truths contain zero information” might be false based on indirect connections between existing work on proof theory and information theory. But I don’t want to give that interpretation yet because I would like to first ask Skatche if he wanted to elaborate on his statement a little better or point to some references for us to read.
The philosophical question is interesting too, and I would agree that a set theory without the axiom of infinity seems pretty adequate for describing our experiences of reality. I’m not sure if its the most harmonious, however...
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I have two different answers to your question: one practical, one more theoretical. On a practical level, what I gain from peak experiences depends on where my attention is. If I’m out and about, or doing something materially, then the main advantage I gain is noticing new aspects of a situation, or seeing the same aspects in a different light; I believe this is a result of greater flexibility in choosing the cognitive map I apply to the territory. These, I suppose, would be the “unknown dots”: information that was present in the environment, but which my brain never bothered to record. If I’m sitting around thinking, on the other hand, then I tend to find a lot of unconventional connections. Even here, though, there is new information to be gained; I’ve learned a lot about my own nervous system simply by careful observation of my internal experience.
On a theoretical level, I have some trouble with the distinction between information and deduction. In the strictest sense, mathematical truths contain zero information, since they are automatically true in every possible world (or insert your own X-Rationalist translation of this claim). Yet we are still surprised to learn, for example, that e^(pi.i)+1=0 - or I know I was surprised, at the very least. I think this is a result of the fact that our evaluation of mathematical claims is based on manipulation of tangible stuff: we can check our memory to determine if we’ve ever seen a proof before, or we can manipulate symbolic expressions, encoded in our wetware, to attempt to prove or disprove it. Even wood pulp and graphite can be leveraged for this purpose, and so the result of this computation comes in as an observation of an uncertain outcome in the world.
Yet the no-information claim takes on new complications when we realize that our brains use the same basic processes to encode relations between facts, as it does to encode facts. This leads to kind of “flat” ontology, in which we can treat relations as facts and draw relations between them, or even between a relation and a fact. We can even draw relations between internal experience (including mathematical knowledge) and external information. Once we have recognized these connections, natural systems can actually point us toward mathematical truths (for example: we might never have learned about solitons if we had not observed them in nature). The hierarchy of “levels of organization”, and the division between information and deduction from information, therefore appear to be constructed post hoc, and I’m not sure if I can cleanly distinguish between unknown dots and unknown relations.
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Thanks, that was an awesome read!
I’ll attempt a translation. If I’m engaging with the world, then I notice new things about it, or I see things in new ways. For example: once, looking at the sky, I noticed that it was brightest near the horizon and darkest at the zenith. Suddenly I realized the reason: there was more air between me and the horizon than there was between me and the space directly above me. The scene snapped into focus, and I found I could distinctly see the atmosphere as a three-dimensional mass. If I turn my attention inward, on the other hand, I tend to draw connections between pieces of information I already have—suddenly intuiting the behaviour of quantum wave packets, for example, or drawing an analogy between my social networking behaviour and annealing. These two cases are the “dots” and the connections between the dots, respectively.
Information theory generally defines information to be about some event with an uncertain outcome: if you know a coin has been flipped, you will need additional information to determine whether it came up heads or tails. By contrast, if you already know that five coins were flipped and three came up heads, you don’t need any additional information to deduce that the number of heads was prime. Anything that you could in principle figure out from the information you’ve already got isn’t treated as new information; In this sense, mathematical truths (connections) are separated from information proper (“dots”).
While this separation may be useful for theory, it doesn’t capture all aspects of the way we learn and process information. For starters, we’re often rather surprised to learn mathematical facts; this is because we need to use physical hardware (our brains) to compute proofs, and we don’t know what the outcome of the computation will be. Also, our brains seem to treat things, states, patterns, and pieces of information all in the same way—hence, for example, we can refer to “the economy” as if it were a single thing rather than a complex system of interrelationships; or, going in the opposite direction, we can break down a tree into its component cells and, moreover, recognize each cell as a fantastically complicated system, and so on.
Meanwhile, our ability to draw links between different levels of organization allows us, in particular, to see that certain mathematical patterns are reflected the world around us. Once we’ve found a model that fits the system, we can make predictions we couldn’t make before: the more confident I am, say, that every fifth coin flip will come up as heads, the less information will be conveyed when this does indeed happen. It goes in the other direction too: sometimes we see patterns in nature which point us toward new mathematical understanding. The example I gave was of soliton waves, which you can read about here—even if you have no technical background, I think you’ll find the History section enlightening.
For all these reasons, I suspect that a better model of information might loosen the hard distinction that’s made between new information and new deductions.
Could you elaborate on that?
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This is a false assertion, they are only true if the axioms used to conclude them correspond to reality. There are proofs that rely on the Axiom of Choice which is not accepted (as far as I can tell) by everyone on this site (as well as the axiom of infinity?). There are proofs that rely on the GCH or Large Cardinal Axioms or V=L which are not among the accepted axioms and proven to be independent of the other axioms.
This is a fair point, but I’m referring to information in the information theoretic sense; in this technical sense, mathematical truths are indeed not information.
I’m aware that the Axiom of Choice is required for some important results of practical import (Tychonoff’s theorem, for example, is equivalent to it), but do you know of any important and useful results following from the GCH, etc.? I’ve only looked into this a little; foundational math is not really my field.
Game theoretic results that are generalized to infinite games often require the use of the GCH. For instance see “Variations on a Game” by J Beck 1981.
It seems Skatche is roughly talking about what is called semantic entailment in logic and I would partially agree with your criticism… since mathematical truth is considered to be more than just that, it includes axioms that you feel good about accepting. However, I’m not sure where reality comes into the picture when considering the definition of mathematical truth.
How do we choose axioms that are good to accept as valid except for our experience with reality? This is a philosophical question so isn’t often considered within mathematics. If one reads some of Eliezer’s posts where he doubts the axiom of infinity (unless I am misunderstanding what he is doing) then it becomes clear that his argument for doubting the axiom is that it doesn’t correspond with reality.
This is generally why choice is doubted; as it is thought that non-measurable sets do not play any role in reality they are ignored in standard statistics and calculus. If non-measurable sets do play a role within the real world then some very odd things can happen that so far no one has observed happen.
I suppose if mathematical truth is changed to be whatever is provable given all possible combinations of non-contradictory axioms then reality does not play any role in math.
I think when trying to study information and its relation to mathematical truth, we must start off practical and should be talking about provability in formal systems of logic. I don’t actually know of any rigorous connections between the two notions, but I can think of an argument that “mathematical truths contain zero information” might be false based on indirect connections between existing work on proof theory and information theory. But I don’t want to give that interpretation yet because I would like to first ask Skatche if he wanted to elaborate on his statement a little better or point to some references for us to read.
The philosophical question is interesting too, and I would agree that a set theory without the axiom of infinity seems pretty adequate for describing our experiences of reality. I’m not sure if its the most harmonious, however...