Consider the mapping between a physical system and its representation. There are degrees of freedom in how the mapping is done. We should like the invariant parts of the respresentation to correspond to invariant parts of the physical system and likewise with variant parts. We’d like the variant parts to vary continuously if they vary continuously in the physical system and likewise for discretely. Some representations are tighter in that they have such type matching along more dimensions. A sparse representation that only captures some of the causal structure of the physical system (lossy) can be desirable if the other dimensions don’t generate externalities relevant to our intent (the representation is modular in the same way that reality appears to be eg chemistry). When we find a lossless representation that has all of its variable parts varying in exactly the same way in the representation we bundle the whole thing up as an equation. That is to say a conservation relation.
This may all sound straightforward, tautological even. But I think it’s worth examining in closer detail what the act of formalization is. Because of course we aren’t actually comparing representations to physical systems, we’re comparing representations with representations. Degrees of invariance is all we have. When we seek a way to test a hypothesis eg whether gavagai refers to a rabbit, a part of a rabbit, or a situation that includes a rabbit, we’re seeking a way to collapse a degree of freedom in the respresentation. Sentences cut down the degrees of freedom in the relation between things until intent is clear. A hypothesis is of the form ‘dimension X appears to vary, but is actually a function of dimension Y’ which decreases the size of the search space by a whole dimension. Words are hypotheses about how reality is bundled. Sentences are hypotheses about how bundles relate.
Consider the mapping between a physical system and its representation. There are degrees of freedom in how the mapping is done. We should like the invariant parts of the respresentation to correspond to invariant parts of the physical system and likewise with variant parts. We’d like the variant parts to vary continuously if they vary continuously in the physical system and likewise for discretely. Some representations are tighter in that they have such type matching along more dimensions. A sparse representation that only captures some of the causal structure of the physical system (lossy) can be desirable if the other dimensions don’t generate externalities relevant to our intent (the representation is modular in the same way that reality appears to be eg chemistry). When we find a lossless representation that has all of its variable parts varying in exactly the same way in the representation we bundle the whole thing up as an equation. That is to say a conservation relation.
This may all sound straightforward, tautological even. But I think it’s worth examining in closer detail what the act of formalization is. Because of course we aren’t actually comparing representations to physical systems, we’re comparing representations with representations. Degrees of invariance is all we have. When we seek a way to test a hypothesis eg whether gavagai refers to a rabbit, a part of a rabbit, or a situation that includes a rabbit, we’re seeking a way to collapse a degree of freedom in the respresentation. Sentences cut down the degrees of freedom in the relation between things until intent is clear. A hypothesis is of the form ‘dimension X appears to vary, but is actually a function of dimension Y’ which decreases the size of the search space by a whole dimension. Words are hypotheses about how reality is bundled. Sentences are hypotheses about how bundles relate.
Thanks, that was a useful way to think of things.