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>Traditional math (which came mostly out of physics and geometry) studies primarily infinite but countable structures (functions on countable sets, infinite ser
by ccortes 6y ago
>Traditional math (which came mostly out of physics and geometry) studies primarily infinite but countable structures (functions on countable sets, infinite series). The finite structures are often considered trivial, and the uncountable structures are considered to be too large to have practical importance.
I don't think this is the case at all. Basically all of calculus theory in based on uncountable structures and it's arguably the most used branch of mathematics.
- js8 6y agoI understand what you're saying, but disagree. In calculus, uncountable sets are just the theater, the real actors are sequences. As long as the theater is a complete metric space, we don't really care about its structure. Hence the name, it signifies that we are unconcerned with cardinality above aleph 0. The finite/countable/uncountable distinction IMHO shows where the focus is. Analogically, in CS, you use infinite sets (integers) but you don't really care what the cardinality is. You don't care about the axiom of choice.
- ccortes 6y ago> As long as the theater is a complete metric space, we don't really care about its structure. I agree with that, but I fail to see how that means that traditional math studies primarily infinite countable structures.
- js8 6y agoPerhaps better would be to say that traditional math studies structures constructed from countable building blocks (and we don't care so much whether the result is countable or uncountable). In contrast with CS, where the building blocks are finite, and the resulting structures can be large finite or infinite, which in practice only matters a little.