I like this mode of thinking, and its angle is something I haven’t considered before. How would you interpret/dissolve the kind of question I posed in the answer to Pattern in the comments below?
Namely:
‘My point is the process of maths is (to a degree) invented or discovered, and under the invented hypothesis, where one would adhere to strict physicalist-style nominalism, the very act of predicting that the solutions to very real problems are dependent on abstract insight is literally incompatible with that position, to the point where seeing it done, even once, forces you to make some drastic ramifications to your own ontological model of the world.’
In addition, I am versed in rudimentary category theory, and wouldn’t a presupposition of having abstraction as natural transformation presuppose the existence of abstraction to define itself? This may be naive of me, or perhaps I have not grasped your subtlety, but using an abstract notion like natural transformation to define abstraction seems circular to me.
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To “dissolve” the math invented/discovered question, it’s a false dichotomy, as constructing mathematical models, conscious or subconscious, is constructing the natural transformations between categories that allow high “compression ratio” of models of the world. They are as much “out there” in the world as the compression would allow. But they are not in some ideal Platonic world separate from the physical one. Not sure if this makes sense.
wouldn’t a presupposition of having abstraction as natural transformation presuppose the existence of abstraction to define itself?
There might be a circularity, but I do not see one. The chain of reasoning is, as above:
1. There is a somewhat predictable world out there
2. There are (surjective) maps from the world to its parts (models)
3. There are commonalities between such maps such that the procedure for constructing one map can be applied to another map.
4, These commonalities, which would correspond to natural transformations in the CT language, are a way to further compress the models.
5. To an embedded agent these commonalities feel like mathematical abstractions.
I do not believe I have used CT to define abstractions, only to meta-model them.
Don’t worry it no trouble :) Thank you, I see your reasoning more clearly now, and my thought of circularity is no longer there for me. Also I see the mental distinction between compression models and platonic abstracts.
I like this mode of thinking, and its angle is something I haven’t considered before. How would you interpret/dissolve the kind of question I posed in the answer to Pattern in the comments below?
Namely:
‘My point is the process of maths is (to a degree) invented or discovered, and under the invented hypothesis, where one would adhere to strict physicalist-style nominalism, the very act of predicting that the solutions to very real problems are dependent on abstract insight is literally incompatible with that position, to the point where seeing it done, even once, forces you to make some drastic ramifications to your own ontological model of the world.’
In addition, I am versed in rudimentary category theory, and wouldn’t a presupposition of having abstraction as natural transformation presuppose the existence of abstraction to define itself? This may be naive of me, or perhaps I have not grasped your subtlety, but using an abstract notion like natural transformation to define abstraction seems circular to me.
Sorry, my spam filter ate your reply notification :(
To “dissolve” the math invented/discovered question, it’s a false dichotomy, as constructing mathematical models, conscious or subconscious, is constructing the natural transformations between categories that allow high “compression ratio” of models of the world. They are as much “out there” in the world as the compression would allow. But they are not in some ideal Platonic world separate from the physical one. Not sure if this makes sense.
There might be a circularity, but I do not see one. The chain of reasoning is, as above:
1. There is a somewhat predictable world out there
2. There are (surjective) maps from the world to its parts (models)
3. There are commonalities between such maps such that the procedure for constructing one map can be applied to another map.
4, These commonalities, which would correspond to natural transformations in the CT language, are a way to further compress the models.
5. To an embedded agent these commonalities feel like mathematical abstractions.
I do not believe I have used CT to define abstractions, only to meta-model them.
Don’t worry it no trouble :) Thank you, I see your reasoning more clearly now, and my thought of circularity is no longer there for me. Also I see the mental distinction between compression models and platonic abstracts.