3 ms·

I don't buy this rant. I found his last argument a bit weak. Just because you can't write a number because it is so big, it doesn't mean it can't be simbolized

I don't buy this rant. I found his last argument a bit weak. Just because you can't write a number because it is so big, it doesn't mean it can't be simbolized by a handful of characters, or abstracted successfully. Also it is more powerful to abstract than it is to enumerate.

ColinWright 15y ago
Here's the thing. Let's suppose you write down all the syntactically valid 1 byte programs (in some language, say, Python). there aren't many of them, and you can certainly write them all down. Now write down all the syntactically valid 2 byte programs, then 3, then 4, and so on. Now you have a list of all valid programs. Some won't terminate, and some won't print anything, but any number that can ever be computed by any program will be printed by (at least) one of these. But there are only countably many of them. That means that most of the numbers between 0 and 1 will not be printed by any program. Ever. Are you now still happy talking about the collection of all the reals? Most of them (by which I mean uncountably many compared with countably many) can never be printed or calculated.
- lobo_tuerto 15y ago
  Well, I'd say then, that's what abstractions are for! Seriously, I think that's part of the role of abstractions, and that's why we _currently_ can talk about and use reals. Even though we can't write the entire Pi number, and the program used to calculate all its digits might never finish, we do have a symbol for it, and put it to good use. What do you think?
  - ColinWright 15y ago
    But we do have a symbol for pi and we can write a program which, for any required N, can print the first N digits. But there are only countably many numbers for which that is true. Uncountably many numbers cannot have symbols and cannot have programs to represent them. If we can never name them, not even in principle, does it make sense even to say that they exist? (Note: Don't think you can extrapolate my own personal view from the above, I am merely pointing out that there are non-trivial questions here, questions that most people don't seem to notice or understand.)

ColinWright 15y ago

Here's the thing. Let's suppose you write down all the syntactically valid 1 byte programs (in some language, say, Python). there aren't many of them, and you can certainly write them all down. Now write down all the syntactically valid 2 byte programs, then 3, then 4, and so on. Now you have a list of all valid programs. Some won't terminate, and some won't print anything, but any number that can ever be computed by any program will be printed by (at least) one of these. But there are only countably many of them. That means that most of the numbers between 0 and 1 will not be printed by any program. Ever. Are you now still happy talking about the collection of all the reals? Most of them (by which I mean uncountably many compared with countably many) can never be printed or calculated.

lobo_tuerto 15y ago
Well, I'd say then, that's what abstractions are for! Seriously, I think that's part of the role of abstractions, and that's why we _currently_ can talk about and use reals. Even though we can't write the entire Pi number, and the program used to calculate all its digits might never finish, we do have a symbol for it, and put it to good use. What do you think?
- ColinWright 15y ago
  But we do have a symbol for pi and we can write a program which, for any required N, can print the first N digits. But there are only countably many numbers for which that is true. Uncountably many numbers cannot have symbols and cannot have programs to represent them. If we can never name them, not even in principle, does it make sense even to say that they exist? (Note: Don't think you can extrapolate my own personal view from the above, I am merely pointing out that there are non-trivial questions here, questions that most people don't seem to notice or understand.)