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I have been thinking about the following problem for years and can't seem to resolve it. I'm looking for more information on the subject and would really appreciate some further reading on the following subject:

Hypothesis: Observations lift reality into the realm of the "Observed reality"; just like mathematics lifts reality into the "logical reality" or "computed reality". In this reality can be seen as "state" and observations, mathematics and even language can be seen as "monads".

By reducing science to a Category of axioms (the objects) and changes (the morphisms), how can we describe "Observations", "Mathematics" and even "Languages" in terms of this Category?

My question is specifically: Is there research being done in this field and are there books/papers I can read on the subject? Thank you in advance for any help/thoughts/opinions/links.

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  • For a start, you'd need to mathematically define, what the structure of an "axiom" object is, such that morphisms in its category preserve that structure; and of course, you'd need to demonstrate that the morphisms compose appropriately.
    – Alexis
    Commented Jun 23, 2016 at 8:54
  • Another thing is, in my experience, 'axioms' in mathematics are typically defined as something fundamental, that is, something that can't be derived from other things. But the existence of a morphism to a supposed "axiom object" implies that object can in fact be derived, which would then make it no longer fundamental.
    – Alexis
    Commented Jun 23, 2016 at 9:00
  • Finally, could you elaborate in what way you think that e.g. observations can be seen as monads in the category-theoretic sense, as distinct from the Leibnizian sense?
    – Alexis
    Commented Jun 23, 2016 at 9:02
  • Thank you for the comments. I can try and answer the questions but keep in mind that I'm not that well versed in CR. I think that within an axiomatic categorical system you can always have derived object. For example, $1$ can be the "axiom" in this case and $+$ a morphism. A "Logical" category would be the natural numbers, "lifting" my axiom and morphism.
    – Baudin999
    Commented Jun 23, 2016 at 9:36
  • Now an "Observational" category would be applying an observation, something done with senses, to an axiom. Imagine being able to lift axiom, for example laws of nature, into "Logical" categories like the category of atoms and again lift this into a "Logical" category of molecules. When I finally observe a rock falling I can create a morphism from a "Logical" category to an "Observational" category by only applying a sense (sight, etc) to the state of my logical system. This sense should have no impact on the underlying structures and can by definition be composed.
    – Baudin999
    Commented Jun 23, 2016 at 9:42

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Possible references:

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  • Thank you for the answer. I'm going to mark this as answered
    – Baudin999
    Commented Jun 27, 2016 at 6:14
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Have you already read the 'Philosophical Significance' section of the SEP's entry on 'Category Theory'?

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