2 ms·

Mostly tangential, but in mathematics, there are sometimes "canonical" objects, typically a 'natural' viewpoint which is proven to be unique and sometimes also

Mostly tangential, but in mathematics, there are sometimes "canonical" objects, typically a 'natural' viewpoint which is proven to be unique and sometimes also 'universal' in a sense. So, in many cases, you can rigorously prove "this is the one 'best' way" (in some sense of best (in some theory)). A top-of-mind example is how a tuple `(A, B)` is "obviously" the (minimal) way you would have both objects A and B within one object: https://en.wikipedia.org/wiki/Product_(category_theory) https://en.wikipedia.org/wiki/Product_(category_theory) I find it interesting that the author mentions "nonscientists", as those seem less likely to be equipped with the experiences of simple/well-defined problems with "global optima". And in mathematics, the "what if there were another way" questions get followed by "suppose there were another way _W_...".

Galanwe 2y ago
> A top-of-mind example is how a tuple `(A, B)` is "obviously" the (minimal) way you would have both objects A and B within one object I am confused as to what that means. A and B are most likely one of the thousands of possible modelisations for an abstract concept I am trying to model. Say I want to model a point on a plane, I could decide to model it as carthesian coordinates in an (A, B) tuple, or a single complex number as a scalar. The _best_ representation depending on how I plan to use that point later on.
- ordu 2y ago
  Complex numbers are just tuples of real numbers with some rules how to add and multiply them.
  - MarkusQ 2y ago
    Not quite. They can be _represented_ that way (but not uniquely, e.g. c=(a+bi) vs abs(c)=r, arg(c)=Θ) or they can be treated as fundamental (defined by axioms), or as a sub-field of the quaternions, etc. If this seems counterintuitive, think of how the claim that "words are just sequences of letters" breaks down when you examine it closely. Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B].
    - ordu 2y ago
      > They can be _represented_ that way (but not uniquely, e.g. c=(a+bi) vs abs(c)=r, arg(c)=Θ) or they can be treated as fundamental (defined by axioms), or as a sub-field of the quaternions, etc. Hmm... If we are looking for the simpliest way, then I don't think that sub-field of the quaternions is simplier than tuples with definitions for operations on them: you need to define quaternions for that. I'm not sure about definition of complex numbers through axioms. Probably it is the simplest way, because for tuples you need first to define real numbers and operations on them. > Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B]. I don't see it either. It is the minimal way to define complex numbers (if we have chosen this path), but why it may be simpler in general case is not obvious. But in any case, I agree, that the whole idea of a "natural" viewpoint was not clearly stated.

Galanwe 2y ago

> A top-of-mind example is how a tuple `(A, B)` is "obviously" the (minimal) way you would have both objects A and B within one object I am confused as to what that means. A and B are most likely one of the thousands of possible modelisations for an abstract concept I am trying to model. Say I want to model a point on a plane, I could decide to model it as carthesian coordinates in an (A, B) tuple, or a single complex number as a scalar. The _best_ representation depending on how I plan to use that point later on.

ordu 2y ago
Complex numbers are just tuples of real numbers with some rules how to add and multiply them.
- MarkusQ 2y ago
  Not quite. They can be _represented_ that way (but not uniquely, e.g. c=(a+bi) vs abs(c)=r, arg(c)=Θ) or they can be treated as fundamental (defined by axioms), or as a sub-field of the quaternions, etc. If this seems counterintuitive, think of how the claim that "words are just sequences of letters" breaks down when you examine it closely. Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B].
  - ordu 2y ago
    > They can be _represented_ that way (but not uniquely, e.g. c=(a+bi) vs abs(c)=r, arg(c)=Θ) or they can be treated as fundamental (defined by axioms), or as a sub-field of the quaternions, etc. Hmm... If we are looking for the simpliest way, then I don't think that sub-field of the quaternions is simplier than tuples with definitions for operations on them: you need to define quaternions for that. I'm not sure about definition of complex numbers through axioms. Probably it is the simplest way, because for tuples you need first to define real numbers and operations on them. > Also, I'm not seeing why a tuple (A,B) (with order) is simpler than a set {A,B} or a bag [A,B]. I don't see it either. It is the minimal way to define complex numbers (if we have chosen this path), but why it may be simpler in general case is not obvious. But in any case, I agree, that the whole idea of a "natural" viewpoint was not clearly stated.