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Naive question: Can you build something like calculus on a number system weaker than the reals, but stronger than the rationals? What's the weakest you could g

Naive question: Can you build something like calculus on a number system weaker than the reals, but stronger than the rationals? What's the weakest you could go?

sverona 7y ago
It's unclear what precisely you mean by 'weaker', but if you mean with regard to completeness - the reals are the so-called completion of (that is, the smallest complete metric space containing) the rationals; so you're kind of stuck.
- liuyao 7y ago
  You can weaken "completeness" though. In each of its many manifestations, completeness asks for the existence of something for all subsets (sequences, functions, ...) of reals. We can instead ask for the ones that are definable by finite means (aka computable, constructive). To find (=construct) the extreme value, for example, you probably won't be able to find a general procedure; it has to be treated case by case. Now you'll appreciate how easy real numbers are. It gives the assurance that the extreme values exist. I would not have thought it would interest Hacker News readers, but here's more about computable numbers: https://observablehq.com/@neobourbaki/computable-numbers https://observablehq.com/@neobourbaki/computable-numbers
  - sverona 7y ago
    This is really cool. My advisor is a strict constructivist and will tell anyone with ears about finitism after a few beers, and I've always found it a useful philosophical position.
tathougies 7y ago
How do you define something 'like' calculus? How do you define 'number' system? If by number system you mean arbitrary ring and by calculus you mean the derivative for example, then yes, but it's not super interesting. For example, the structure of algebraic data types forms a ring, and you can define an operation 'like' the derivative on it. However, given that only two operations (multiplication and addition) are commonly defined here, it's not particularly involved, although the result itself is quite useful (the derivative of the algebraic form of a data type is the type of one-holed versions of that same data type).

Naive question: Can you build something like calculus on a number system weaker than the reals, but stronger than the rationals? What's the weakest you could go?

sverona 7y ago

It's unclear what precisely you mean by 'weaker', but if you mean with regard to completeness - the reals are the so-called completion of (that is, the smallest complete metric space containing) the rationals; so you're kind of stuck.

liuyao 7y ago
You can weaken "completeness" though. In each of its many manifestations, completeness asks for the existence of something for all subsets (sequences, functions, ...) of reals. We can instead ask for the ones that are definable by finite means (aka computable, constructive). To find (=construct) the extreme value, for example, you probably won't be able to find a general procedure; it has to be treated case by case. Now you'll appreciate how easy real numbers are. It gives the assurance that the extreme values exist. I would not have thought it would interest Hacker News readers, but here's more about computable numbers: https://observablehq.com/@neobourbaki/computable-numbers https://observablehq.com/@neobourbaki/computable-numbers
- sverona 7y ago
  This is really cool. My advisor is a strict constructivist and will tell anyone with ears about finitism after a few beers, and I've always found it a useful philosophical position.

tathougies 7y ago

How do you define something 'like' calculus? How do you define 'number' system? If by number system you mean arbitrary ring and by calculus you mean the derivative for example, then yes, but it's not super interesting. For example, the structure of algebraic data types forms a ring, and you can define an operation 'like' the derivative on it. However, given that only two operations (multiplication and addition) are commonly defined here, it's not particularly involved, although the result itself is quite useful (the derivative of the algebraic form of a data type is the type of one-holed versions of that same data type).