3 ms·

I assume that you mean solutions where x is in R here

by beebmam 5y ago

mjburgess 5y ago
And in my view, `R` is "fundamentally" a geometrical construct.
- zarzavat 5y ago
  R is an analytical construct, it exists to make limits meaningful. You don't need all of R to do geometry, if you are Ancient Greek you don't need any field at all.
- contravariant 5y ago
  It's the only complete ordered field, so fundamentally it's an algebraic structure (with some topology thrown in but basic enough that it can be phrased without referring to geometry).
  - mjburgess 5y ago
    It depends what you give "priority" to -- my view is a philosophical one. I think we ought really see `R` as an algebraic stand-in for talking about geometry.
    - contravariant 5y ago
      Eh, if R is geometric then so is algebra. The only thing geometric about R is that you can define an absolute value (i.e. it is an ordered ring) which is a property it inherits from the fractions. The fractions in turn are a basic consequence of the definition of a field (the fractions have a unique image into all fields). If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.
  - rocqua 5y ago
    "complete" here is a matter of analysis / topology much more than it is a matter of algebra. In some sense, the algebraic numbers are indeed important in algebra as being the 'basic' field with infinite characteristic. You get to `R` by demanding completeness (for others reading, the demand that every converging sequence has an existing limit). I would say that the idea of limits and completeness are fundamentally topological / analytical. Which is closer to geometry than to algebra. I think the main conclusion here is that there is good reason that the field "algebraic geometry" exists.
    - contravariant 5y ago
      General topology is a pretty far cry from geometry though. What you need for the reals isn't even directly related to the topology of R, since you just need that every set with an upper bound has a least upper bound, which is a property that many types of orders can have and which doesn't necessarily have anything to do with geometry.
rssoconnor 5y ago
I do mean that in the sense that you only need ℝ. However, it is also the case that the solutions over the complex numbers also all have period 2π.

I assume that you mean solutions where x is in R here

mjburgess 5y ago

And in my view, `R` is "fundamentally" a geometrical construct.

zarzavat 5y ago
R is an analytical construct, it exists to make limits meaningful. You don't need all of R to do geometry, if you are Ancient Greek you don't need any field at all.
contravariant 5y ago
It's the only complete ordered field, so fundamentally it's an algebraic structure (with some topology thrown in but basic enough that it can be phrased without referring to geometry).
- mjburgess 5y ago
  It depends what you give "priority" to -- my view is a philosophical one. I think we ought really see `R` as an algebraic stand-in for talking about geometry.
  - contravariant 5y ago
    Eh, if R is geometric then so is algebra. The only thing geometric about R is that you can define an absolute value (i.e. it is an ordered ring) which is a property it inherits from the fractions. The fractions in turn are a basic consequence of the definition of a field (the fractions have a unique image into all fields). If you've got addition and multiplication then you can get the reals by just adding additive inverse, multiplicative inverses and completeness, none of which are inherently geometric properties. So any geometry must come from the distributive property, because there's not much else it could be.
- rocqua 5y ago
  "complete" here is a matter of analysis / topology much more than it is a matter of algebra. In some sense, the algebraic numbers are indeed important in algebra as being the 'basic' field with infinite characteristic. You get to `R` by demanding completeness (for others reading, the demand that every converging sequence has an existing limit). I would say that the idea of limits and completeness are fundamentally topological / analytical. Which is closer to geometry than to algebra. I think the main conclusion here is that there is good reason that the field "algebraic geometry" exists.
  - contravariant 5y ago
    General topology is a pretty far cry from geometry though. What you need for the reals isn't even directly related to the topology of R, since you just need that every set with an upper bound has a least upper bound, which is a property that many types of orders can have and which doesn't necessarily have anything to do with geometry.

rssoconnor 5y ago

I do mean that in the sense that you only need ℝ. However, it is also the case that the solutions over the complex numbers also all have period 2π.