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A big difference is that standard analysis is well-defined on its own. The standard Copenhagen interpretation is simply not a well-defined theory because measur
by jostylr 10y ago
A big difference is that standard analysis is well-defined on its own. The standard Copenhagen interpretation is simply not a well-defined theory because measurement is not well-defined, e.g., the cat problem.
A more appropriate analogy to the SA v NSA would be GRW formulation vs Bohmian or even a well-defined many worlds (that does exist, but it does not involve splitting of worlds which is as problematic as collapse). These are different theories and they can lead to different generalizations.
In particular, GRW seems more amenable to being relativistic without adding a foliation or some other structure.
Bohmiam mechanics, on the other hand, does requires using a foliation though there are possibilities to use within the existing structures:
https://arxiv.org/abs/1307.1714 https://arxiv.org/abs/1307.1714
I also think NSA has a naming problem. By its very name, how can it ever be standard?
I think the issue is also that the limit fits in with numerical analysis in that it talks about errors. NSA sounds like it skips the error analysis and is only concerned with stuff at the limit which is certainly useful in many instances, but not always.
- MichaelBurge 10y ago> I also think NSA has a naming problem. By its very name, how can it ever be standard? "Nonstandard" isn't referring to how people view the field, but to additional elements you add to the model. Just to throw out a weird example, here's Edward Nelson's text on the subject: https://web.math.princeton.edu/~nelson/books/1.pdf https://web.math.princeton.edu/~nelson/books/1.pdf "Theorem 4. There is a finite set that contains every standard object." That is, in Nelson's system, the set of standard natural numbers is finite. That finite number is a nonstandard natural number, larger than any standard natural number. Mixing "nonstandard"(in the sense of an undefined predicate added to the base logic) with "nonstandard"(in the sense of some professors thinking it's weird) is a type error.