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No, but you may have to put aside some preconceptions to get the point. https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy https://en.wikipedia.
by btilly 5y ago
No, but you may have to put aside some preconceptions to get the point.
https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_controversy https://en.wikipedia.org/wiki/Brouwer%E2%80%93Hilbert_contro... is a good starting point on the historic debate over the philosophy of math.
From within Formalism, which for all intents and purposes won, your characterization is correct. We have the standard reals. And we've constructed a proper subset of the reals.
From within Constructivism, the "standard reals" is a piece of sophistry. It is ridiculous to claim the existence inconceivable infinite swarms of non-existent things whose only claim to "existence" is the sheer multitude of numbers that can never be named or constructed. And therefore the "reals" that I described are a sensible thing to call reals, all of whom have an existence that can be established on reasonable grounds.
From within either philosophy, the other doesn't make much sense. But, in fact, both philosophies are internally consistent, and no logical argument can ever establish one over the other. (In fact, Formalism won because it is more convenient. And for no other reason.)
- wtallis 5y ago> you may have to put aside some preconceptions to get the point. "Preconceptions" is a rather unhelpful way to describe the actual situation here, which is you using a different definition for real numbers, that is not even remotely equivalent, without making that clear from the outset. You said: > But there is a countable list that DOES include every possible real - you just can't always figure out whether things on that list are reals! Which is pretty clearly incorrect using the usual definition of real numbers. The fact that you don't like how real numbers is usually defined is not sufficient justification for you to start confusing a discussion by mixing in your alternative definition without making an explicit distinction. You've defined a set that can certainly be the subject of interesting investigation, even in a context where the usual definitions about real numbers and uncountable infinities are still accepted. Choosing to inject a naming conflict is counterproductive and suggests you're more concerned with making smug claims about being able to do things mainstream mathematicians consider impossible, rather than having a productive discussion about how to construct most of familiar mathematics without allowing uncountable infinities.
- btilly 5y agoThe name "real numbers" predates the first formal definitions. The philosophical debate that I pointed to predates the general acceptance of the standardization of the modern definition of the reals. Now standard definitions literally makes no sense within constructivism. You talk about constructing familiar mathematics, but are using non-constructions that depend on questionable and questioned notions of absolute truth. What I described is as close to standard mathematics as you can come within a constructivist framework. The tradition of calling such constructions "the real numbers" may be new to you, but is actually over a century old.