Meta-meta-meta-meta....language
It is interesting how functional constraints imposed by the mere form of objects of interest can give rise to novel characteristics. Yet, language sets the upper bound on what one can communicate about novel characteristics. And what makes a characteristic novel anyway? If I flip a quarter a million times, any particular outcome (when considering the entire string of flips) will be highly unlikely to observe. But we could care less if the information encoded by the string of 1's and 0's is truly random. What we truly care about is if there is some deeper pattern or organization to the string of 1's and 0's. But given an apparently random string of 1's and 0's how can we tell if it is truly random? What is randomness? This is a deep question in computer science relating to the most efficient algorithm to produce a particular output (e.g. a string of "random" 1's and 0's).
It seems likely that in the space of possible string's of 1's and 0's representations of ourselves and our reality(ies) exist in an infinite multitude of forms. This is both terrifying and awe-inspiring. In fact I would argue that if power-sets of infinities (ad nauseum) exist, then reality is much weirder than we could even begin to imagine, i.e. the holes and bridges that actually exist across all relations and non-relations are infinitely more complex than we can even begin to begin to...understand.
How does one go about constructing increasingly powerful languages, that can identify all (or at least most) salient characteristics of objects which they operate on? There is an upper bound on how powerful a language can be (i.e. it can't evaluate the truth of its own statements in finite time). But that doesn't mean we can't build languages that more accurately mirror the form of the objects they operate on. Say we have some complicated web of causal interactions with noise (protein folding, gene networks, membrane dynamics, any complicated ensemble process...). We want to talk in meaningful ways about the system. All we have to do is identify all conditional independencies in state-space, then all meta-conditional independencies (e.g. conditional independencies between groups of variables), then all meta-meta...(groups of groups), etc... These may also be multilinear groups of conditional independencies, (i.e. we may have hyper-(n)-graphs). This will parse the system into the relevant hierarchies. From this highly complex hyper-graph, we can develop qualitative state-space analysis techniques, essentially cutting up the state space based on the structure of conditional independence. Every level of conditional independency corresponds to some higher order structure we cannot see merely at the bottom level, but which may play an important role in the dynamics/behavior of the system. The most interesting cases will be where the meta-* conditional independencies end up feeding back down to the bottom of the hierarchy (and vice versa), thereby partitioning state-space in increasingly complex ways. (Feedback can be used to break symmetries in state space, including symmetries on higher order structures.) We can then assign probability to entire chunks of state space based on the inferred global structure of the system in question. This probability assignment will allow us to make statements about the current and future state of the system, couched in the noisy nature of our understanding of the system in question. We can then operate on this hypergraph embedded in some nonlinear manifold. We can ask questions like, if I couple these subsystems, what will happen to the global dynamics of my system? Since we know all the conditional independencies, we can immediately identify which structures any operations will affect, and then it should be possible to focus on just those structures to elaborate what the effects will be.
To make each level of meta-conditional independency tractable, there will need be clever notational and semantic rules. A language constructed in this framework would have the advantage of being both general and specific, which means that one could taylor the language well for multiple systems, then compare each of these specific languages to look for isomorphism. More on this when I'm not crunched for time...
It seems likely that in the space of possible string's of 1's and 0's representations of ourselves and our reality(ies) exist in an infinite multitude of forms. This is both terrifying and awe-inspiring. In fact I would argue that if power-sets of infinities (ad nauseum) exist, then reality is much weirder than we could even begin to imagine, i.e. the holes and bridges that actually exist across all relations and non-relations are infinitely more complex than we can even begin to begin to...understand.
How does one go about constructing increasingly powerful languages, that can identify all (or at least most) salient characteristics of objects which they operate on? There is an upper bound on how powerful a language can be (i.e. it can't evaluate the truth of its own statements in finite time). But that doesn't mean we can't build languages that more accurately mirror the form of the objects they operate on. Say we have some complicated web of causal interactions with noise (protein folding, gene networks, membrane dynamics, any complicated ensemble process...). We want to talk in meaningful ways about the system. All we have to do is identify all conditional independencies in state-space, then all meta-conditional independencies (e.g. conditional independencies between groups of variables), then all meta-meta...(groups of groups), etc... These may also be multilinear groups of conditional independencies, (i.e. we may have hyper-(n)-graphs). This will parse the system into the relevant hierarchies. From this highly complex hyper-graph, we can develop qualitative state-space analysis techniques, essentially cutting up the state space based on the structure of conditional independence. Every level of conditional independency corresponds to some higher order structure we cannot see merely at the bottom level, but which may play an important role in the dynamics/behavior of the system. The most interesting cases will be where the meta-* conditional independencies end up feeding back down to the bottom of the hierarchy (and vice versa), thereby partitioning state-space in increasingly complex ways. (Feedback can be used to break symmetries in state space, including symmetries on higher order structures.) We can then assign probability to entire chunks of state space based on the inferred global structure of the system in question. This probability assignment will allow us to make statements about the current and future state of the system, couched in the noisy nature of our understanding of the system in question. We can then operate on this hypergraph embedded in some nonlinear manifold. We can ask questions like, if I couple these subsystems, what will happen to the global dynamics of my system? Since we know all the conditional independencies, we can immediately identify which structures any operations will affect, and then it should be possible to focus on just those structures to elaborate what the effects will be.
To make each level of meta-conditional independency tractable, there will need be clever notational and semantic rules. A language constructed in this framework would have the advantage of being both general and specific, which means that one could taylor the language well for multiple systems, then compare each of these specific languages to look for isomorphism. More on this when I'm not crunched for time...
posted by meta_darwinist at 8:23 PM
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