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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

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.

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.