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> schools are always going to teach and require the status quo. Rephrase this: "Schools are always going to teach and require the mathematics that most people
by derleth 13y ago
> schools are always going to teach and require the status quo.
Rephrase this: "Schools are always going to teach and require the mathematics that most people use, especially given that most people are going into fields where math is a tool, not an end in itself."
A different example of this is how Nonstandard Analysis remains 'nonstandard' even though it was essentially how both Newton and Leibniz originally imagined calculus: Until Robinson showed how to put infinitesimals on a rigorous footing, the only axiomitization of calculus involved epsilon-delta proofs and so that became, and still remains, the standard form of analysis.
Now, most people learning calculus will never need to know how it came to be, but axioms give a structure to guide thinking, and that is why they're taught to non-mathematicians. The precise axioms which are chosen is of somewhat secondary importance compared to the habit of mind formed by internalizing some axiom system and learning to think on that basis at least some of the time.
- thaumasiotes 13y agoNote that nonstandard analysis is formalized by a constructive equivalence to 'standard' analysis; it's not a good example of the idea "other maths are possible if we agree with different rules", since there is no difference anywhere (except potentially in the mental model, where NSA is superior -- but all the math is the same). For "other maths are possible if we agree with different rules" I always think of euclidean vs noneuclidean geometry, where you actually get different results. But that's not a good fit with saying the reals are uncountable; as far as I know people who are unhappy with different infinite cardinalities don't have an alternate system (and in fact cannot have an alternate system, since the diagonalization theorem is a theorem where the parallel postulate is an axiom); they just deal with their unhappiness by ignoring the idea.
- kriro 13y agoAnd often it's quite useful to understand the "standard" to see why deviations were thought up. An example would be non-classical logic.