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You might want to be more explicit about what you mean by "everything" and "represented". :-) There's the conventional point that things other than abstract structure and abstract relationships can't, in some sense, be represented by mathematics at all. You can make a mathematical model that is isomorphic to some part of reality (including cultural and social details) but one can argue that what we mean by all kinds of human concepts relates to concrete, not abstract, things and experiences, which are probably not representable in mathematics. That doesn't just include "love" or "beauty" but also things like "Earth", "Barack Obama", "French", "professor", "Internet", or "Mother's Day". Those isomorphisms can sometimes be very precise and very useful, which is why we can have, for example, physics simulations (which are based on isomorphisms between a mathematical model and a physical situation, which can be strong and precise enough to make detailed and useful predictions about how such a physical situation will evolve). Some people believe that reality itself is mathematical at its root, in which case all of these things actually are either ill-defined (like it's not possible to guarantee that objects are or are not included in the category) or are mathematical objects, but we would still not expect to have access to those mathematical theories themselves because they would be below the level of our reality. Apart from that, mathematical theories have limitations such as incompleteness (where consistent theories are typically not able to answer all questions that can be posed in them). Just thinking about mathematical research, I presume that there are mathematical structures which we could represent with existing mathematics, but which we haven't realized are interesting or important yet, so we don't have a habit of doing so and don't have an established vocabulary for those. Like groups prior to the development of group theory! |