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Around the turn of the last century, the logicists, like Frege and Russell, attempted to reduce all of mathematics to logic; to prove that all mathematical truths were logical truths. However, the systems they used (provably) failed, because they were inconsistent.
Furthermore, it seems likely that any attempt at logicism must fail. Firstly, any system of standard mathematics requires the existence of an infinite number of numbers, but modern logic generally has very weak ontological commitments: they only require the existence of a single object. For mathematics to be purely logical, it must be tautological - true in every possible world*, and yet any system of arithmetic will be false in a world with a finite number of elements.
Secondly, both attempts to treat numbers as objects (Frege) or concepts/classes (Russell) have problems. Frege’s awful arguments for numbers being objects notwithstanding, he has trouble with the Julius Caesar Objection; he can’t show that the number four isn’t Julius Caesar, because what this (abstract) object is is quite under-defined. Using classes for numbers might be worse; on both their systems, classes form a strict hierarchy, with a nth level classes falling under (n+1)th classes, and no other. Numbers are defined as being the concept which has all those concept’s whose elements are equinumerous; the class of all pairs, the class of all triples, etc. But because of the stratification, the class of all pairs of objects is different from the class of all pairs of first level classes, which is different from the class of all pairs of second level classes, and so on. As such, you have an infinite number of ‘2’s, with no mathematical relations between them. Worse, you can’t count a set like {blue chair, red chair, truth, justice}, because it contains objects and concepts.
What seems more likely to me is that there are an infinite variety of mathematical structures, purely syntax without any semantic relevance to the physical world, and without ‘existence’ in any real sense, as a matter of induction we’ve realised that some can be interpreted in manners relevant to the external world. As evidence, consider the fact that different, mathematics are applicable in areas: probability theory here, complex integration here, addition here, geometry here...
*strictly speaking, true in every structure.
I appreciate your comments but I'm having trouble seeing your point with regards to the idea. To reiterate, with regards to your last paragraph,
I'm proposing that these interpretations work because the internal physical systems (the territory) obeys the same properties as consistent mathematical systems -- see my comment to TM below.