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>Should these be called numbers or rather just processes which arrive at numbers? They are one and the same. You can use a Graham's number place value system a

by zardo 9y ago

>Should these be called numbers or rather just processes which arrive at numbers? They are one and the same. You can use a Graham's number place value system and Graham's number is simply written while a number like 7 might be quite complex to represent.

pmoriarty 9y ago
If a number and the process used to arrive at a number were completely equivalent, then going through the process to arrive at the number would never be worthwhile. Yet people are often interested in actually going through these processes to generate numbers, when it's practical to do so.
- zardo 9y ago
  That's because they want to represent the number in a particular representation. They are not "completely" equivalent, they are equivalent up to the representation, converting between representations can have very high computational costs. But equivalent up to the representation is the usual notion of equality in mathematics, and if you mean something else you have to define it. (Obviously in computer science, this difference is quite important and we need to differentiate between an expression and it's evaluation)
  - pmoriarty 9y ago
    There still seems to be a difference in kind between a number represented in such a way that no further processing is needed, versus represented in a way in which further processing is needed. To me, such a fundamental difference seems to deserve a distinction in nomenclature.
    - Retra 9y ago
      That's a distinction in goals though, not in concepts. If we want to know how many people are on a bus "the number of people you would count if you bothered" is a perfectly true answer. It's an accurate representation. You perhaps can't do anything more than talk vaguely about abstract bus populations with it, but that is a reasonable thing to do. On the other hand, maybe "17" is the answer you want, because you need to compare it to some other integer or plug it into a function. Also useful things to do, but not always. And not even more often than the trivial alternatives. Even though we're significantly biased toward wanting the latter form, it still means the same thing as the former. And the former is a description of a process, while the latter is a number. Or the former is a number, while the latter is a particular encoding of a process. Same difference in the end. We even define numbers by describing the processes we use to generate them.

by zardo 9y ago

pmoriarty 9y ago

If a number and the process used to arrive at a number were completely equivalent, then going through the process to arrive at the number would never be worthwhile. Yet people are often interested in actually going through these processes to generate numbers, when it's practical to do so.

zardo 9y ago
That's because they want to represent the number in a particular representation. They are not "completely" equivalent, they are equivalent up to the representation, converting between representations can have very high computational costs. But equivalent up to the representation is the usual notion of equality in mathematics, and if you mean something else you have to define it. (Obviously in computer science, this difference is quite important and we need to differentiate between an expression and it's evaluation)
- pmoriarty 9y ago
  There still seems to be a difference in kind between a number represented in such a way that no further processing is needed, versus represented in a way in which further processing is needed. To me, such a fundamental difference seems to deserve a distinction in nomenclature.
  - Retra 9y ago
    That's a distinction in goals though, not in concepts. If we want to know how many people are on a bus "the number of people you would count if you bothered" is a perfectly true answer. It's an accurate representation. You perhaps can't do anything more than talk vaguely about abstract bus populations with it, but that is a reasonable thing to do. On the other hand, maybe "17" is the answer you want, because you need to compare it to some other integer or plug it into a function. Also useful things to do, but not always. And not even more often than the trivial alternatives. Even though we're significantly biased toward wanting the latter form, it still means the same thing as the former. And the former is a description of a process, while the latter is a number. Or the former is a number, while the latter is a particular encoding of a process. Same difference in the end. We even define numbers by describing the processes we use to generate them.