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Aside

This is a silly observation born out of my own curiosity. I'd love some perspective from more mathematically minded folks.

Please try to evaluate my reasoning, and expand on which assumptions are incorrect or misguided.

General question

Why are points considered to be zero dimensional? Is there a distinction between zero dimensional and dimensionless that I am missing?

Assumptions

  • A single point cannot exist in the absence of at least one dimension.
  • The point has no meaning beyond its distance from zero along that dimension.

Conclusion

  • A single point is not dimensionless.
  • Every point is coupled to some dimension(s), and its meaning is generated via its distance to zero along those dimension(s).
  • In this sense, every point has content, as it assumes a value on a dimension. A dimensionless point would have no capacity to assume a value.
efreezy
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    What does "dimension" mean to you, exactly? And "A single point cannot exist in the absence of at least one dimension" is like saying that a line cannot exist in the absence of at least a plane. Of course it can! Finally, "The point has no meaning beyond its distance from zero along that dimension" doesn't describe the point, it describes the ambient space. – Arthur Aug 23 '18 at 06:51
  • You can define the special vector space that only has the point ${0}$ in it. It has dimension 0. – ippiki-ookami Aug 23 '18 at 06:53
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    You are "playing" with concepts that are not clearly defined and your logics lacks rigor. We cannot follow you. For instance, you assume that we understand what you mean by "zero dimensional" vs. "dimensionless". How could we ? –  Aug 23 '18 at 06:58
  • Yves, thanks for the critique. I implore you to encourage those who lack rigor in their formal reasoning, and to strive to understand and improve their mental model. Especially those who indeed have genuine curiosity. A more community-oriented, socially productive rephrasing of your response might read "I am happy to see you playing with these concepts, however I think you would benefit by taking a step back and articulating more clearly what you mean by "zero dimensional" vs "dimensionless" – efreezy Aug 25 '18 at 20:02

1 Answers1

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Here's just one viewpoint.

You may be interested in the difference between affine dimension and linear dimension. For flat things:

  1. The affine dimension is the least number of points needed to specify the thing.
  2. The linear dimension is the number of orthogonal directions available to someone living on the thing.

For example, since there's a unique line through any two points, ergo a line has affine dimension $2$. But if you're living on a line, there's only one direction along which you can move back and forth, so it has linear dimension $1$.

In general, the linear dimension is always $1$ less than the affine dimension.

Since a point takes $1$ point to specify, its affine dimension is $1$. Thus its linear dimension is $0$. Since "dimension" usually means "linear dimension", hence a point can be considered $0$-dimensional.

goblin GONE
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