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The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of somet

by j2kun 12y ago

The question is, "What do you define as a 'problem'?" You need to answer that before you can make claims like "we can all agree..." and give an example of something that I don't agree is a problem (if you're saying what I think you're saying).

JadeNB 12y ago
My weak, but (I think) practical, implicit definition of 'problem' was: > a question that a mathematician might ask I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.) By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)
- j2kun 12y ago
  You're falling into the interesting number paradox here, regardless of not using the word interesting. You can't make a statement about the cardinality of a set if your definition of that set isn't expressed using mathematics. In other words, you're saying "I'm not going to give a mathematical definition of this thing, but I can nevertheless give a mathematical proof of a mathematical statement about that thing." For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!

by j2kun 12y ago

JadeNB 12y ago

My weak, but (I think) practical, implicit definition of 'problem' was: > a question that a mathematician might ask I agree that this is not a very useful definition, by virtue of its extreme and probably excessive inclusiveness, but I think that it's hard to do any better without using words like 'interesting' that themselves need careful definition. (It's fair to argue that my definition in turn requires clarification of the term 'mathematician', but I can weasel my way around that by replacing it with 'person', or else just declaring that anyone interested in trying to ascertain the normality of a number is mathematically minded enough to be called a mathematician.) By this definition, I think that it is hard to argue with my claim to have produced an uncountable family of problems—simply because, at least classically, to do so you'd have to produce a specific number x about whose normality no mathematician could ever ask. I could then demolish that counterexample by asking you if that particular number x was normal. :-)

j2kun 12y ago
You're falling into the interesting number paradox here, regardless of not using the word interesting. You can't make a statement about the cardinality of a set if your definition of that set isn't expressed using mathematics. In other words, you're saying "I'm not going to give a mathematical definition of this thing, but I can nevertheless give a mathematical proof of a mathematical statement about that thing." For example, why is the set of all questions utterable by humans not a countable set? By that reasoning there are more numbers than questions!