What's the difference between Euclidean & Cartesian space?

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Discussion Overview

The discussion revolves around the differences between Euclidean space and Cartesian space, exploring definitions, dimensions, and the relationship between these concepts. It includes theoretical considerations and clarifications of terminology.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants assert that there is no difference between Euclidean and Cartesian space, suggesting that "Cartesian space" may be a misnomer for Cartesian coordinates within Euclidean spaces.
  • Others argue that Cartesian space refers specifically to two or three dimensions with mutually perpendicular axes, while Euclidean space can extend to higher dimensions.
  • One participant emphasizes that Euclidean space is defined by satisfying Euclid's axioms, whereas Cartesian space is described as a set of ordered pairs of real numbers, implying a connection to rectangular coordinates.
  • There is a suggestion that confusion arises from the conflation of terms, particularly regarding the use of "Cartesian space" versus "Cartesian coordinates."

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of Euclidean and Cartesian spaces, indicating that multiple competing interpretations remain unresolved.

Contextual Notes

Some definitions and assumptions about dimensions and coordinate systems are not fully explored, leaving room for ambiguity in the terms used.

swampwiz
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What's the difference between Euclidean & Cartesian space?
 
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swampwiz said:
What's the difference between Euclidean & Cartesian space?
One exists, the other doesn't.
One doesn't speak of Cartesian spaces. What you mean is probably Euclidean spaces and thus there is no difference, but only because you invented a term. One speaks of Cartesian coordinates in Euclidean spaces, which means the coordinate directions are pairwise perpendicular. Euclidean space mean, there is no curvature. E.g. the surface of the moon is curved and so no Euclidean space. The screen on which I read this now is flat, and thus Euclidean.
 
swampwiz said:
What's the difference between Euclidean & Cartesian space?
I've never heard the term "Cartesian space," but if I search for it on the web, I find some hits. More often I see "Cartesian coordinates."

From one of the definitions I saw, a Cartesian space is one of either two or three dimensions, in which the axes are mutually perpendicular.

A Euclidean space also has mutually perpendicular axes, but can represent spaces of higher than three dimensions.
 
Most likely authors are conflating the terms of Cartesian space to mean Cartesian coordinates in a Euclidean space.
 
jedishrfu said:
Most likely authors are conflating the terms of Cartesian space to mean Cartesian coordinates in a Euclidean space.
Better than what had happened to me here on PF. I innocently abbreviated orthonormal system ...
 
A Euclidean space is geometric space satisfying Euclid's axioms. A Cartesian space is the set of all ordered pairs of real numbers e.g. a Euclidean space with rectangular coordinates.
 

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