Is there a category theory of models that would unite theories from different academic disciplines?
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https://en.wikipedia.org/wiki/Category_theory is an abstract mathematical theory that is used to formalize https://en.wikipedia.org/wiki/Mathematics and its concepts as a collection of objects and arrows (also called https://en.wikipedia.org/wiki/Morphism). Category theory can be used to formalize concepts of other high-level https://en.wikipedia.org/wiki/Abstractions such as https://en.wikipedia.org/wiki/Set_theory, https://en.wikipedia.org/wiki/Ring_theory, and https://en.wikipedia.org/wiki/Group_theory.
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Answer:
Interesting question, one possible option is Model Theory which is defined below. It started as a study of languages and has moved into a mainstream model for mathematics. http://plato.stanford.edu/entries/model-theory/
Hunter McCord at Quora Visit the source
Other answers
What does it mean to unify in this context? Are you aware that different sciences use different methods and because of that use different mathematics, and even in mathematics itself category theory has limited applications? That said, there are several branches outside pure mathematics where category theory found its applications. For example take theoretical CS, where it's used in several places, for example in domain theory (which aims at providing mathematical foundations to one approach to semantics of programming languages). It also helps understanding Curry-Howard isomorphism (or maybe rather is part of the correspondence). Some wizards from UK even made IO for purely functional programming language Haskell using monads, which come from category theory (this I think can be thought of as a part of algebraification programme).
Kuba Bartczuk
Though Category Theory has a reputation as difficult (and too abstract) there is some movement towards exploring it as a unifying framework for understanding the relationships between different theoretical areas that currently have their own home-grown mathematics. This would start out by exploring functional analogies but might one day become a common mathematical foundation. As a non-mathematician I wonder whether this could provide a basic theory of models in place of the current use of set theory. The following article explores this idea: http://scholar.google.com/scholar?q=related%3APgH55oF8WggJ%3Ascholar.google.com%2F&hl=en&as_sdt=0%2C48&as_ylo=2005&as_yhi=2015 In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology. Namely, a linear operator behaves very much like a âcobordismâ: a manifold representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and âquantum topologyâ. But this was just the beginning: similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of âclosed symmetric monoidal categoryâ. We assume no prior knowledge of category theory, proof theory or computer science. ... By now there is an extensive network of interlocking analogies between physics, topology, logic and computer science. They suggest that research in the area of common overlap is actually trying to build a new science: a general science of systems and processes. Building this science will be very difficult. There are good reasons for this, but also bad ones. One bad reason is that different fields use different terminology and notation. The original Rosetta Stone, created in 196 BC, contains versions of the same text in three languages: demotic Egyptian, hieroglyphic script and classical Greek. Its rediscovery by Napoleonâs soldiers let modern Egyptologists decipher the hieroglyphs. Eventually this led to a vast increase in our understanding of Egyptian culture. At present, the deductive systems in mathematical logic look like hieroglyphs to most physicists. Similarly, quantum field theory is Greek to most computer scientists, and so on. So, there is a need for a new Rosetta Stone to aid researchers attempting to translate between fields.
Jeff Wright
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