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To answer your first question: Yes. I mean, both. (And no, I'm not offended by your choice of words.) Since you've restated the discussion on more neutral t

To answer your first question: Yes. I mean, both. (And no, I'm not offended by your choice of words.) Since you've restated the discussion on more neutral terms, allow me to rephrase my position in your terms: The article is about how people have trouble swallowing Cantor's argument. Yet clearly Cantor's argument is correct within a framework that includes the reals. And my position is that the existence of Cantor's argument leads me to question the framework itself. You say that there really isn't anything to agree or disagree with, that we are just talking about definitions. Absolutely, I agree. It's unfortunate that the only words we have to talk about these things also have extra connotations that might be more forceful than what we really mean. We're really just talking about the consequences of manipulating symbols under certain rules. HOWEVER... Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and "read off" an interpretable meaning from the answer? At that point may you not make a judgement that the combination of the system you choose and your interpretation of it lead to "believable" results? Is it entirely correct to say that there is no such thing as "agreeing" or "disagreeing" with a definition of a mathematical object, if putting it into the system leads to results you don't consider believable?

jerf 17y ago
"Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and 'read off' an interpretable meaning from the answer?' Ah, and therein lies the rub. In math terminology, I think that that itself is an axiom, in the sense that you can choose it or not choose it, and it will strongly affect the rest of the system. If you accept the axiom, you end up going down the constructivist path. If you do not, you end up in "conventional mathematics" (i.e., what you get for a real math degree in college). Personally, as a programmer I am also sympathetic to the constructivist viewpoint, but I am happy to settle for everyone involved understanding when they've left the physical universe (real numbers, axiom of choice, etc) and when they haven't. If some people want to screw around in the unreal universe, more power to them, doesn't have to bother me. And they often produce useful approximations for real-life use. As RiderOfGiraffes said, there are proofs that work on reals much better than any actual number, and I observe that in practice the resulting maths work out pretty well in the real world, even if they aren't actually physical. (We have way larger problems dealing with floating point, monsters of our own creation, than we do in dealing with the fact that our equations of motion are defined based on real numbers that don't actually correspond to anything, where the inaccuracies are typically beyond our measuring ability and dominated by general inaccuracies of the input data which are far larger than the granularity of the universe.) But I do think maintaining that distinction is definitely important. I further observe that while I can't speak for all of mathematics, the mathematicians are very careful to distinguish what takes the Axiom of Choice and what does not, just to draw one example. They are pretty good about showing their work. I guess the takeaway is that you certainly aren't alone; many people agree that real numbers are ridiculous in the way you are sensing. But they can be a useful fiction.
RiderOfGiraffes 17y ago
> Isn't the whole point of math that you expect at > some point to analyze the result of the symbol > manipulation and "read off" an interpretable > meaning from the answer? I've responded to this at greater length here: http://news.ycombinator.com/item?id=1071734 http://news.ycombinator.com/item?id=1071734 You are absolutely right when it comes to the application of existing math, but in Pure Math that doesn't turn out to be the case.

jerf 17y ago

"Isn't the whole point of math that you expect at some point to analyze the result of the symbol manipulation and 'read off' an interpretable meaning from the answer?' Ah, and therein lies the rub. In math terminology, I think that that itself is an axiom, in the sense that you can choose it or not choose it, and it will strongly affect the rest of the system. If you accept the axiom, you end up going down the constructivist path. If you do not, you end up in "conventional mathematics" (i.e., what you get for a real math degree in college). Personally, as a programmer I am also sympathetic to the constructivist viewpoint, but I am happy to settle for everyone involved understanding when they've left the physical universe (real numbers, axiom of choice, etc) and when they haven't. If some people want to screw around in the unreal universe, more power to them, doesn't have to bother me. And they often produce useful approximations for real-life use. As RiderOfGiraffes said, there are proofs that work on reals much better than any actual number, and I observe that in practice the resulting maths work out pretty well in the real world, even if they aren't actually physical. (We have way larger problems dealing with floating point, monsters of our own creation, than we do in dealing with the fact that our equations of motion are defined based on real numbers that don't actually correspond to anything, where the inaccuracies are typically beyond our measuring ability and dominated by general inaccuracies of the input data which are far larger than the granularity of the universe.) But I do think maintaining that distinction is definitely important. I further observe that while I can't speak for all of mathematics, the mathematicians are very careful to distinguish what takes the Axiom of Choice and what does not, just to draw one example. They are pretty good about showing their work. I guess the takeaway is that you certainly aren't alone; many people agree that real numbers are ridiculous in the way you are sensing. But they can be a useful fiction.

RiderOfGiraffes 17y ago

> Isn't the whole point of math that you expect at > some point to analyze the result of the symbol > manipulation and "read off" an interpretable > meaning from the answer? I've responded to this at greater length here: http://news.ycombinator.com/item?id=1071734 http://news.ycombinator.com/item?id=1071734 You are absolutely right when it comes to the application of existing math, but in Pure Math that doesn't turn out to be the case.